FEA Coursework Report (Sven Cumner 11011794)

FEA Coursework Report

Laminate Analysis and Design and Analysis of a Composite Pressure Vessel

Student: Sven Cumner

Student Number: 11011794

Module Title: Modelling and Simulation (UFMFWD-30-M)

Module Leader: Ramin Amali

Award: MEng Motorsport Engineering

Number of pages: 34

Submission Date: 11/12/2014

Sven Cumner (UFMFWD-30-M) – FEA Coursework 11011794

i

Contents

List of Figures .......................................................................................................................................... ii

List of Tables .......................................................................................................................................... iii

1. Introduction ........................................................................................................................................ 1

1.1. What are fibre-reinforced composite materials? ........................................................................ 1

1.2. Why use composites for pressure vessels? ................................................................................. 1

2. Filament Winding ........................................................................................................................... 2-3

2.1. What is Filament Winding? .......................................................................................................... 2

2.2. Determining composite layup limitations ................................................................................. 2-3

3. Determining Pressure Vessel Resultant Forces Per Unit Length ...................................................... 3

4. Excel Spreadsheet Analysis ............................................................................................................. 3-7

4.1. Designing the spreadsheet - Overview ........................................................................................ 4

4.2. Using Excel Solver to optimize fibre directions and layer thicknesses ..................................... 4-7

5. Abaqus Finite Element Analysis .................................................................................................... 7-20

5.1 Modelling the Pressure Vessel and performing the FEA ......................................................... 7-13

5.1.1. Creating the Part (Part Module) ...................................................................................... 8

5.1.2. Creating the Material and the Composite Layup (Property Module) ......................... 9-10

5.1.3. Creating the Assembly (Assembly Module) ................................................................... 10

5.1.4. Creating the Steps (Steps Module) ................................................................................ 10

5.1.5. Creating the boundary conditions and the internal pressure (Load Module) .......... 11-12

5.1.6.Creating the Mesh (Mesh Module) ................................................................................ 12

5.1.7. Creating the Job (Job Module) ....................................................................................... 12

5.1.8. Visualising the Contour Plot on the Deformed Model (Visualization Module) ............. 13

5.2. Mesh Convergence Study ..................................................................................................... 13-18

5.3. Design Iterations ........................................................................................................................ 18

5.4. Altering the Concrete Support Positions .............................................................................. 18-20

6. Connecting Pipework to Inlets and Outlets ..................................................................................... 21

7. Considering the Weight of the Vessel ........................................................................................ 22-23

8. Consideration of Environmental Temperature Change ............................................................. 24-26

9. Conclusions and Discussion ............................................................................................................. 27

Bibliography.......................................................................................................................................... 28

Appendices ...................................................................................................................................... 29-30


ii

List of Figures

Figure 1 - Typical wet filament winding machinery set up. (Nuplex Industries Ltd., 2014) ................... 2

Figure 2 – Symmetric and balanced laminate layup. Symmetric: due to ply directions being mirrored

about the mid-plane. Balanced: because all theta angle plies have a corresponding negative theta

angle elsewhere in the laminate. (unknown, unknown) ........................................................................ 2

Figure 3 - Screenshot of Excel Solver parameters for optimisation of fibre direction pattern.

Optimised angles can be seen on the right of the image. ...................................................................... 5

Figure 4 - Screenshot of description of angle optimisation values used for Solver. .............................. 5

Figure 5 - Screenshot of Solver parameters used to optimise layer thicknesses (left) and optimised

layer thicknesses (right). ......................................................................................................................... 6

Figure 6 - Pressure vessel form and dimensions (sourced from the project brief:

FEA_Composite_Materials_Coursework_2014-15) ................................................................................ 7

Figure 7 - Initial 2D sketch of the pressure vessel geometry in Abaqus. ................................................ 8

Figure 8 - Basic shell revolution .............................................................................................................. 8

Figure 9 - Shell revolution with holes, partitioned face, and additional datum planes ......................... 8

Figure 10 - Screenshot of Composite Layup data used in Abaqus .......................................................... 9

Figure 11 - Ply stack plot query output ................................................................................................. 10

Figure 12 - Screenshots of Step parameters ......................................................................................... 10

Figure 13 – Edges on the Y-Z plane chosen for the X-axis symmetry constraint .................................. 11

Figure 14 - All boundary condition edge selections .............................................................................. 11

Figure 15 - Global Seeds parameters and boundary nodes represented on vessel to the right .......... 12

Figure 16 - The initial mesh applied to the Pressure_Vessel part ........................................................ 12

Figure 17 - S11 Stress contours plotted on the deformed pressure vessel .......................................... 13

Figure 18 - Coarse mesh (AGS=80) ........................................................................................................ 14

Figure 19 - Normal mesh (AGS=40) ....................................................................................................... 14

Figure 20 - Fine mesh (AGS=20) ............................................................................................................ 14

Figure 21 - Very fine mesh (AGS=10) .................................................................................................... 14

Figure 22 - Nodes selected for mesh study data are highlighted in red. .............................................. 15

Figure 23 – Corner node S11 stress values for varying mesh densities. ............................................... 16

Figure 24 - Corner node S22 stress values for varying mesh densities. ................................................ 17

Figure 25 - Corner node S12 stress values for varying mesh densities. ................................................ 17

Figure 26 - Screenshot images of each deformed vessels with differing concrete support boundary

conditions. D1600 is top, D1500 middle, and D1400 on bottom. Contour plots are showing S11

element for ply 8. .................................................................................................................................. 19

Figure 27 - Very fine mesh around centre hole on D1400 model - an area of potentially infinite stress

concentration. ....................................................................................................................................... 20

Figure 28 - Carbon fibre filament wound pressure vessel (cutaway) with metallic liner featuring

integral pipe bosses (Composites World, 2012) ................................................................................... 21

Figure 29 – Side view of pressure vessels showing effect of negative (top) and positive (bottom)

gravity. Negative gravity models the top half of the vessel correctly, whereas positive gravity models

the bottom half of the vessel correctly................................................................................................. 22

Figure 30 - The positive/negative gravity vessels superimposed on to each other to create

representative image of (exaggerated) realistic deformation .............................................................. 23

Figure 31 - Screenshots of expansion coefficient additions to composite material properties ........... 24

Figure 32 - Screenshot of Predefined Field settings to represent initial temperature reference. ....... 25


iii

List of Tables

Table 1 - Mesh convergence data for corner node............................................................................... 15

Table 2 - Mesh convergence data for left end of cap base ................................................................... 16

Table 3 - Mesh convergence data for right end of cap base ................................................................ 16

1

1. Introduction

1.1. What are fibre-reinforced composite materials?

Composite materials are made up of two or more significantly different constituent materials, that

come together to form a material with different characteristics than those of the individual

components. Fibre-reinforced composite materials are obtained by embedding fibres of a strong,

stiff material into a weaker, softer material, referred to as a matrix. Typical materials used as fibres

are carbon, glass, and polymers, while various types of resins are typically used as a matrix. A layer,

ply, or lamina, of a composite material consists of a large number of parallel fibres embedded in a

matrix. (Ferdinand P. Beer, 2012)

An axial load applied to a lamina along the x axis (the axis parallel to the fibre direction) will create a

normal stress (S11) in the lamina and a corresponding normal strain (ε11) which will satisfy Hooke’s

law as the load is increased and as long as the elastic limit of the lamina is not exceeded. Similarly,

an axial load applied along the y axis (the axis perpendicular, in plane, to the fibre direction) will

create a normal stress (S22) and a normal strain (ε22) satisfying Hooke’s law, and an axial load

applied along the z axis (perpendicular, out of plane, to fibre direction) will create a normal stress

(S33) and a normal strain (ε33) which again satisfy Hooke’s law. However, the moduli of elasticity E1,

E2, and E3 corresponding, respectively, to each of the above loadings will be different. As the fibres

are parallel to the x axis, the lamina will offer a much stronger resistance to a loading directed along

the x axis than to a loading directed along the y or z axis, and E1 will be much larger than either E2 or

E3. Materials whose properties are dependent on the direction considered are said to be

anisotropic. There will also be shearing stresses generated, the one of most importance (for this

study) being the S12 component, which is the shearing stress of a particular element of the lamina,

parallel to the XY plane, as a result of any load acted upon the lamina. (Ferdinand P. Beer, 2012)

1.2. Why use composites for pressure vessels?

Working with composite materials offers engineers the chance to build stronger, lighter structures,

when compared to traditional materials, due to their higher strength-to-weight ratios. They also

allow engineers to take advantage of their anisotropic properties, giving the ability design a

structure that is strong only in the directions it needs to be.

“Cylindrical pressure vessels with integrated end domes develop hoop stresses that are

twice longitudinal stresses and when isotropic materials, like metals, are used, the material

is not fully utilized in the longitudinal direction, resulting in over weight components.”

(M.Madhavi, 2009).

This indicates that using an ordinary isotropic material (properties same in all directions), such as

steel, would be wasteful due to the material being able to unnecessarily take twice the pressure

resultant loading in the longitudinal direction.


2

2. Filament Winding

2.1. What is Filament Winding?

Filament winding consists of winding a continuous roving of fibre onto a rotating mandrel in

predetermined patterns. This method of manufacturing provides the greatest control over fibre

placement and uniformity of structure. In the wet winding method, the fibre picks up matrix

material, such as resin, either by passing through a matrix bath or from a metered application

system. In the dry winding method, the matrix is in the pre-impregnated form, known as tow-preg.

Refer to fig.1 below to view a typical wet winding machinery setup. After the desired layers have

been wound, the component is cured and removed from the mandrel (or in some cases the mandrel

becomes part of the component). Filament winding is traditionally used to produce pressure vessels,

pipe, rocket motor casings, tanks, ducting, golf club shafts and other symmetric parts. (McClean

Anderson, 2014)

The filament winding process will be controlled by software on a control computer. The process

program needs to be carefully designed to avoid excessive gaps and excessive overlaps between

each winding band, whilst minimising the winding circuits needed for each band. Cylindrical pressure

vessels also have a special consideration in that the winding pattern on the end caps needs to be

carefully designed to avoid excessive material build up due to the bands overlapping each other at

the peaks of the hemi-spherical end caps.

2.2. Determining composite layup limitations

Figure 2 – Symmetric and balanced laminate layup. Symmetric: due to ply directions being mirrored about the mid-plane. Balanced: because all theta angle plies have a corresponding negative theta angle elsewhere in the laminate. (unknown, unknown)

Figure 1 - Typical wet filament winding machinery set up. (Nuplex Industries Ltd., 2014)


3

A symmetrical and balanced laminate (as in fig.2 above) is critical to prevent internal stresses and

deformation due to uneven thermal expansion during the curing process. Consecutive plies with

more than 45 degrees difference in orientation should be avoided. This minimizes inter-laminar

shear effect and prevents manufacturing deformations. (Amali, 2014)

3. Determining Pressure Vessel Resultant Forces Per Unit Length

As mentioned in section 1.2, cylindrical pressure vessels develop hoop stresses that are twice the

magnitude of the longitudinal stresses. This suggests the relationship is the same between hoop and

longitudinal resultant forces due to internal pressure. This is indeed the case; the equations for

calculating these resultant forces (per unit length) are as follows:

where Nx = resultant force per unit length in longitudinal direction, Ny = resultant force per unit

length in hoop direction, P = internal pressure, and R = radius of vessel curvature. There are no

external loads/torques applied to the pressure vessel, hence there are no resultant shear forces and

no resultant moments; Nxy = Mx = My = Mz = 0. (M.Madhavi, 2009)

The given pressure value is 60Bar = 6MPa = 6*106 N/m2. The radius applicable to the author’s

pressure vessel analysis is 400mm = 0.4m. Hence, the force calculations are as follows:

So the resultant forces are Nx = 1.2*106 N/m, Ny = 2.4*106 N/m.

4. Excel Spreadsheet Analysis

4.1. Designing the spreadsheet - Overview

(Please see corresponding file, “Composite Laminate Spreadsheet – Sven Cumner - 11011794”).

An Excel spreadsheet workbook was designed, by the author, to allow optimisation of certain

aspects of a composite laminate material using complex numerical simulations. Specifically, the

optimisations of lamina ply thicknesses, angles of ply direction in the layup, and various Factors of

Safety (FoS). The spreadsheet is set up to allow optimisation of any laminate with a number of plies

between 1 and 20, and can take in to consideration the effects of temperature change and moisture

absorption. There are three main worksheets (coloured with red tabs) that are used for the ‘Part C’

analysis. (Note: Blue-tab sheets are the initial simplified worksheets for Parts A and B, created as part

of the development process, working towards the more advanced red-tab sheets.)

Analysis begins on the input worksheet, where all of the various input parameters can be

entered. These include the material properties, stress limits, resultant forces/moments due

to internal pressure, ply angles/thicknesses, and the number of plies in the laminate.


4

The matrix worksheet contains all of the various matrices required to compute the global

stress values, which in turn are used in the FoS sheet to determine local stress values and

then the various FoS values for each ply of the laminate.

The FoS worksheet contains the calculations required to determine the FoS value dependent

on maximum allowable compressive/tensile stresses, and the FoS values due to the Tsai-Hill

and Tsai-Wu failure theories. The minimum FoS value for each of these three critera is

determined as the respective minimum FoS for the entire laminate. These three final

minimum FoS values are also output on the inputs page, to allow the user to see the results

of experimentation with differing input values, without having to switch worksheets.

The spreadsheet will check to make sure the minimum FoS for the laminate is above 1.2 (minimum

FoS value as stipulated by the project brief), and will then display the FoS values on the input page

with a green or red background fill colour to indicate whether all values are over 1.2 or not.

However, it is useful to note that the Tsai-Wu criterion has consistently given the lowest FoS value

out of the three criteria.

The spreadsheet also checks to see if the vessel is thin-wall. This is important to check, as some of

the mathematics used in the spreadsheet are only applicable to thin-wall pressure vessels. A

pressure vessel is considered to be thin-walled if its radius, R, is larger than 5 times its wall thickness,

tk. (eFunda, 2014). This also helps to minimise the weight of the vessel.

Unless otherwise stated, all mathematical principles used in the spreadsheet were taken from week

9 to week 13 lecture notes of ‘UFMFWD-30-M Modelling and Simulation’ module lectures on ‘FEA of

Composite Materials’, held at the University of the West of England, Bristol, from September to

October 2014.

There are also several further ‘yellow-tab’ worksheets beyond the Part A, B, and C section sheets.

These were created for the collection of Abaqus FEA stress data collection, as discussed later in this

report (please refer to sections 5, 7 and 8).

4.2. Using Excel Solver to optimise fibre directions and layer thicknesses

‘Solver’ is a Data Analysis add-in feature of the Excel software program that allows optimisation of

an objective value, by changing one or more variable values, subject to a series of constraints. Solver

was first used to determine the optimum fibre directions (known together as the layup) and then,

the minimum required layer thicknesses to maintain a Tsai-Wu FoS of higher than 1.2 (through initial

experimentation, it was found that the Tsai-Wu criterion always gave the lowest FoS value). The

layer thicknesses were kept equal for all layers, to simplify the analysis and create a more

realistically efficient layup, in terms of manufacturing capabilities. All following screenshots were

taken from, and during the analysis of, the Excel spreadsheet file “Composite Laminate Spreadsheet

– Sven Cumner – 11011794”.


5

4.2.1. Fibre direction optimisation using Solver

Fig.3 above shows the parameters used to optimise the fibre direction layup pattern. These

parameters are described below.

The objective value was the Tsai-Wu FoS value, which was set to be solved for its maximum

possible value. At this stage, all layer thicknesses were set to 1mm (a rough estimate of a

suitable layer thickness).

The variable cells were the first four

layer angles. The last four angles

were set to be symmetric to the first

four, hence only the first four were

required to be varied, as the last four

would automatically appear

symmetric to the varied values.

The first constraint value relates to a

table of more complicated functions

that were required to give an overall

output value of 1 when the layer

angles were of a suitable layup

pattern. More information on this

table can be found in fig.4 to the

right. (Note: the process could have

Figure 3 - Screenshot of Excel Solver parameters for optimisation of fibre direction pattern. Optimised angles can be seen on the right of the image.

Figure 4 - Screenshot of description of angle optimisation values used for Solver.


6

been streamlined by only using the top three values in the table shown in fig.4, as the bottom

three are symmetric to the top three anyway)

The second and third constraint values simply ensure that only angles between -90 and 90

degrees were chosen, to match with standard laminate layup reference standards. (Note:

values between 0 and 180 degrees are also typically seen in laminate layup patterns, where,

for example, a value of 135 degrees is equal to an angle of -45 degrees.)

The optimum angles can be seen in the fig.3 screenshot on the previous page. The laminate code for

this layup is [-45/0/45/90]S. Please note that the actual angles output by the solver were not integers

(the first angle was -45.49983275, for example), so these angles were rounded to their nearest

whole number to achieve the more realistically reproducible angles shown.

4.2.2 Layer thickness optimisation using Solver

Fig.5 below shows the Solver parameters used to optimise the layer thicknesses, along with the

optimised thickness values. The parameters used in Solver are described below.

The objective value is ‘t’, the overall thickness of the laminate, which was set to be solved

for its minimum value.

The variable cell was just set to change the value for t1, as all other thicknesses were set to

equal the first layer thickness, so would automatically be set equal to this first value.

The first constraint ensures that the Tsai-Wu FoS remains greater than, or equal, to 1.2 - the

minimum allowable FoS as stipulated by the project brief.

The second and third constraints give the thickness values a reasonable range to be set

within. 0.001m (1mm) was known to be an approximately suitable thickness, so a range

between 0.01m (10mm) and 0.0001m (0.1mm) was used to keep the Solver calculations to a

realistic timescale. It was very unlikely that any optimum value would be outside this range.

Figure 5 - Screenshot of Solver parameters used to optimise layer thicknesses (left) and optimised layer thicknesses (right).


7

The optimum layer thickness value can be seen in fig.5. Again, this value was actually rounded up

from an approximate value of 1.215mm, to give a value with a more realistically suitable number of

significant figures. To summarise, each layer was determined to have an optimum thickness of

1.22mm, giving a total laminate thickness of 9.76mm. At this wall thickness, the pressure vessel is

still well within the constraints of being considered ‘thin wall’.

5. Abaqus Finite Element Analysis

SIMULIA Abaqus FEA is a software suite for Finite Element Analysis and Computer-Aided

Engineering. Abaqus/CAE is the program used to perform the FEA. Abaqus was used to design, and

analyse, a CAD model representing one eighth of the pressure vessel. Only creating an eighth of the

vessel may seem peculiar, but was done to save computational processing power/time when the

model was analysed. Abaqus has the ability to create symmetry constraint boundary conditions, to

simply mirror the results of the eighth-model simulation in the X, Y, and Z planes to effectively

simulate a whole pressure vessel just from the original eighth. The process used to create and

analyse this model is detailed in the following sub-sections.

5.1. Modelling the Pressure Vessel and performing the FEA

(Please refer to supporting Abaqus file “Pressure_Vessel_Sven_Cumner_11011794.cae”)

The following process was based upon the tutorial guide, “Introductory Analysis of a Fibre

Reinforced Composite Material”, as found on the UWE Blackboard file storage network. It was

created and delivered by David Fisher as part of the UFMFWD-30-M Modelling and Simulation

module at the University of the West of England, Bristol.

The final goal of this section was to model a pressure vessel to match the project brief specification

drawings, shown below in fig.6. There were two inlets, one on each hemi-spherical end cap of the

pressure vessel and two outlets on the top and bottom of cylindrical centre section. The diameters

of the inlets and outlets were given in the drawings to be 60mm. The pressure vessel was to be

supported by two concrete supports; these were assumed to be rigid compared to the pressure

vessel. Analysis of alternate concrete support positions is documented in section 5.4 of this report,

but for the main analysis they are assumed to be 1600mm apart, as per the diagram. The radius of

curvature of the vessel, applicable specifically to the author’s analysis, was 400mm.

The pressure vessel was subject to an internal pressure of 60bar.

Figure 6 - Pressure vessel form and dimensions (sourced from the project brief: FEA_Composite_Materials_Coursework_2014-15)


8

Figure 8 - Basic shell revolution

Figure 9 - Shell revolution with holes, partitioned face, and additional datum planes

Please note that all following values used in Abaqus are with respect to the units of Newtons (N)

and millimetres (mm or m*10-3). Also note the primary model

described here was named “D1600”.

5.1.1. Creating the Part (Part Module)

After Abaqus had been opened, the ‘Create Part’ icon was

selected. In the ‘Create Part’ dialog box, the following

parameters were selected:

Name: “Pressure_Vessel”

Modelling Space: 3D

Type: Deformable

Base Feature Shape: Shell

Base Feature Type: Revolution

Approximate size: 3000

In the 2D section sketching window, the sketch shown in fig.7 to

the right was created. This basic shape has a straight vertical

right edge of 1000mm, and a 400mm radius curve at the top.

This sketch represents a cross section of an eighth of the whole

vessel, which would have a cylindrical centre section 2000mm

long, with a 400mm radius hemi-spherical end cap on each end.

The next dialog box was the ‘Edit Revolution’

window, in which an angle of 90 degrees was

specified to revolve the model around the

axis of the left-hand vertical construction line in the

sketch shown in fig7. The basic 3D shell revolution

was then created, as can be seen in fig.8 to the left.

The next step was to create the holes in the

top/bottom of the vessel and the holes in the end

caps, to represent the pipe inlets and outlets. To do

this, two additional datum planes were created,

one 1400mm offset from the X-Z plane, and one

offset 400mm from the X-Y plane. On these planes,

the appropriate one-quarter circles were sketched, and then these

shapes cut-extruded through the relative shell faces to create the

quarter holes in the appropriate positions on the eighth model.

The main cylindrical face was then partitioned 800mm up from its

base, to create a line on that face that could later be chosen for the

boundary condition to represent the concrete supports. The model

with partitioned face and cut-extruded holes can be seen in fig.9 to the

right, along with the two offset datum planes (in yellow).

Figure 7 - Initial 2D sketch of the pressure vessel geometry in Abaqus.


9

5.1.2. Creating the Material and the Composite Layup (Property Module)

The first step here was to create the material that the pressure vessel will be made out of. The

‘Create Material’ icon was selected, the material was named “Composite”, and it was given the

properties as listed in the project brief. These properties were entered in to the ‘Edit Material’ dialog

box, as listed below (along with some steps on how to get to the relevant fields):

General>Density>Mass Density = 1.45e-6 kg/mm3 (equal to the given value 1450kg/m3)

For Mechanical>Elasticity>Elastic, ‘Lamina’ was chosen from the ‘Type’ drop-down list.

Longitudinal Modulus, E1 = 120,000 N/mm2

Transverse Modulus, E2 = 8,000 N/mm2

Poisson’s Ratio, ν12 = 0.3

Shear Modulus, G12 = 6,000 N/mm2



Note: The two shear moduli G13 and G23 were approximated from the ratio of the values used in

the tutorial example (delivered by Dave Fisher during Modelling and Simulation tutorial sessions)

that indicated a G12 value 20% larger than the others, where G12 was 3,600 and G13, G23 = 3,000.

The next step was to create the ‘Composite Layup’, which was named “Pressure_Vessel_Layup”. The

‘Element Type’ for the layup was set to ‘Conventional Shell’, and the ‘Initial Ply Count’ was set to 8.

The relevant parameters were then filled out as shown below in fig.10, to match the thickness and

orientation values obtained from the Excel Solver optimisation (refer to section 4.2 for these values).

The ‘(Picked)’ regions represent the whole part, shown highlighted in purple on the right. The

‘Normal Direction’ gave reference to each ply for the desired direction of the global axes.

Figure 10 - Screenshot of Composite Layup data used in Abaqus


10

Figure 12 - Screenshots of Step parameters

The composite layup was visually checked for correctness by going to ‘Query’ in the ‘Tools’ menu,

selecting ‘Ply stack plot’ from the ‘Property Module Queries’ pop-up menu, and selecting any face on

the vessel model. The results of this query are shown below in fig.11.

5.1.3. Creating the Assembly (Assembly

Module)

As this model is just one part, the instance

type is set to Dependent. The ‘Create

Instance’ dialog box was brought up and ‘OK’

was simply clicked to complete this section.

The model turned blue to indicate it had been

placed in to an assembly.

5.1.4 Creating the Steps (Step Module)

In the ‘Create Step’ dialog box, a new step

was named “Pressurise”, the procedure type

was set to ‘General’, and ‘Static, General’ was

chosen from the list. The ‘Edit Step’ dialog

box was set up as shown in fig.12 to the right.

Figure 11 - Ply stack plot query output


11

5.1.5 Creating the boundary conditions and the internal pressure load (Load Module)

As mentioned earlier, symmetry constraints were required to reflect the results of the eighth model

analysis and effectively simulate an entire pressure vessel. The data entered in to the ‘Create

Boundary Condition’ dialog boxes to create the symmetry constraints was as follows:

Names for each symmetry: “X_Symm_BC”, “Y_Symm_BC”, and “Z_Symm_BC”

Step: Initial

Category: Mechanical

Types for Selected Step: Symmetry/Antisymmetry/Encastre

In the ‘Edit Boundary Condition’ dialog box,

the corresponding radio button was

highlighted for each symmetry constraint (i.e.

for the X-axis symmetry, XSYMM

(U1=UR2=UR3=0) was selected).

For each of the symmetry constraints, the

model edges situated on the plane

perpendicular to that axis of symmetry were

selected. For example, the model edges lying

on the Y-Z plane were selected for the X-axis

symmetry constraint, as can be seen in fig.13

to the right.

Once all of the symmetry constraints had

been created, it was necessary to create fixed boundary

conditions to simulate the concrete supports and the rigid

pipe connections. This was done by first entering the same

values as for the symmetry constraints in to the ‘Create

Boundary Condition’ dialog box, with condition names of

“Support” and “Solid Pipe Connections”. The edge previously

created 800mm up the main face (by partitioning it in section

5.1.1) was selected for the Support condition. The arcs

representing the pipe connection holes were selected for the

Solid Pipe Connection condition. In the ‘Edit Boundary

Condition’ dialog box, ENCASTRE

(U1=U2=U3=UR1=UR2=UR3=0) was chosen for both support

and pipe conditions, to ensure that the specified edges could

not move in any global direction – they were fixed in space.

All of the above boundary condition edge selections can be

seen in fig.14 to the right.

Figure 13 – Edges on the Y-Z plane chosen for the X-axis symmetry constraint

Figure 14 - All boundary condition edge selections


12

The next step was to define the internal pressure load condition. This was done by first selecting the

‘Create Load’ icon. In the ‘Create Load’ dialog box, the following data was input:

Name: “Apply_Pressure”

Step: Pressurise

Category: Mechanical

Types for Selected Step: Pressure

All faces of the pressure vessel were then selected, and the appropriate colour (brown in this case)

was chosen to represent the internal shell face of the vessel.

In the ‘Edit Load’ dialog box, the only modification made was to set the magnitude to 6. This

represents a value of 6N/mm2, which is equivalent to the 60bar pressure stated in the project brief.

5.1.6 Creating the Mesh (Mesh Module)

The first step in creating the mesh was

to select the ‘Assign Mesh Controls’

icon. In the ‘Mesh Controls’ dialog box,

the Element Shape was set to ‘Quad’

and the Technique was set to

‘Structured’.

Next, the ‘Assign Element Type’ icon

was selected. In the ‘Element Type’

dialog box, the ‘Standard’ Element

Library was selected, along with setting

the ‘Family’ to ‘Shell’.

The ‘Seed Part’ icon was then selected,

and the given parameters (with

approximate global size of 40) shown in

fig.15 to the right were accepted.

The ‘Mesh Part’ icon was then selected, and the mesh shown in

fig.16 to the right was then generated.

5.1.7 Creating the Job (Job Module)

The ‘Create Job’ icon was selected, and in the ‘Create Job’ dialog

box the suitable model (named “D1600”, in reference to the

concrete support positions) was selected. The job name was set

as ‘2-D1600’ as a previous test job had been named ‘1-Test-Job’.

No modifications were made in the ‘Edit Job’ dialog box.

The job was then submitted and the program left to carry out its

necessary calculations. The results were visualised and collected

in the next step.

Figure 15 - Global Seeds parameters and boundary nodes represented on vessel to the right

Figure 16 - The initial mesh applied to the Pressure_Vessel part


13

Figure 17 - S11 Stress contours plotted on the deformed pressure vessel - all mirror planes activated.

5.1.8 Visualising the Contour Plot on the Deformed Model (Visualization Module)

The ‘Plot Contours on Deformed Shape’ icon was selected; this allows various output variable

distribution plots to be viewed over the (exaggerated) deformed pressure vessel surface. To view the

entire vessel as a whole part, not just the one-eighth section, the following command sequence was

performed: View>OBD Display Options>Mirror/Pattern. This brought up a dialog box in which the

‘Mirror planes’ (‘XY’, ‘XZ’, ‘YZ’) could be activated via their respective radio buttons.

The S11 stress component for Ply-1 gave the contours/deformed model shown below in fig.17.

5.2. Mesh Convergence Study

When creating an FEA model, it is important to develop a sufficiently refined mesh to ensure that

the results from the Abaqus simulation are adequate. Coarse meshes can yield inaccurate results,

but the computer resources required to run the simulation will be comparatively small; as the mesh

gets finer, the simulation will require more and more computer resources. The numerical solution

provided by the model will generally tend toward a unique value as mesh density is increased. The

mesh is said to be converged when further mesh refinement produces a negligible change in the

desired solution. (Abaqus, unknown)

Working in the Mesh module again, selecting the ‘Seed Part’ icon brought up the ‘Global Seeds’

dialog box, within which the ‘Approximate global size’ was set to an initial value of 80. This

produced the coarse mesh shown in fig.18 on the next page. The approximate global size was then

also set to 40 (fig.19), 20 (fig.20), 15, 10 (fig.21), and 9. A global size of 8 gave a number of nodes

beyond the maximum limit (20,000) allowable in the student version of Abaqus; hence a global size

of 9 provided the finest mesh possible in this situation.


14

For all of these global sizes, the S11, S22, and S12 stress values were collected for three nodes at

separate areas of the model, all of which retained consistent geometry coordinates. These nodes

were:

The node positioned at the lower right corner of the eighth model. This is the centre node

on each side of the centre portion of the whole vessel, with an axis perpendicular to the axes

of the pipe holes (i.e. the node at the centre where the ‘bold cross’ can be seen in the fine

meshes above). Global co-ordinates of (400, 0, 0).

The node on the left end of the line defining the transition from the vertical cylinder surface

to the hemispherical cap surface. Global co-ordinates of (0, 1000, 400).

The node on the right end of the line described above. Global co-ordinates of (400, 1000, 0).

None of these nodes move position when a finer mesh is used, due to their co-ordinates being on

key points of geometry definition (i.e. these nodes are essential to define the intrinsic shape of the

vessel). These nodes described above, are shown highlighted in red in fig.22 below.

Figure 21 - Very fine mesh (AGS=10) (6695 elements) Figure 20 - Fine mesh (AGS=20) (1565 elements)

Figure 18 - Normal mesh (AGS=40) (390 elements) Figure 19 - Coarse mesh (AGS=80) (107 elements)


15

Along with logging the aforementioned stress values, the time taken for each simulation to complete

was also noted. The data sets for each node are shown below in Table 1, Table 2, and Table 3.

Table 1 - Mesh convergence data for lower right hand corner node.

Global size S11 S22 S12 Time (min)

80 41.5393 2.97683 3.46713 00:07

40 40.6618 3.1788 3.55913 00:20

20 39.9209 3.20731 3.5672 00:39

15 39.5308 3.21224 3.56979 00:50

10 39.1641 3.21244 3.57086 01:22

9 39.1565 3.20651 3.57144 01:11

Figure 22- Nodes selected for mesh study data are highlighted in red.


16

Table 2 - Mesh convergence data for left end of cap base


80 38.3822 3.53628 0.00129 00:08

40 38.1421 3.60328 0.133602 00:20

20 37.1277 3.53364 0.265822 00:45

15 36.8019 3.51333 0.304674 00:57

10 36.4648 3.4873 0.346337 01:26

9 36.4011 3.4825 0.354849 01:35

Table 3 - Mesh convergence data for right end of cap base


80 37.9436 8.23426 -8.0622 00:07

40 38.6799 9.07331 -8.50631 00:19

20 39.9976 9.36899 -8.61284 00:46

15 40.3417 9.96429 -8.92903 00:48

10 41.1774 10.4999 -9.24371 01:21

9 41.3315 10.7279 -9.3693 01:31

From the data shown in each table above, nine graphs were plotted in Excel to view the trends of

each stress element at each node during the convergence study. The graphs for the corner node are

shown below and the graphs for the cap base ends are shown in Appendix A. Fig.23 shows the graph

for S11 elements, fig.24 shows the graph for S22 elements, and fig.25 shows the graph for S12

elements. Figs.A1-A6 show the graphs for S11, S22, S12, for the two end cap base nodes.

37.5

38

38.5

39

39.5

40

40.5

41

41.5

42

80 40 20 15 10 9

Stre

ss (

N/m

m2 )

Approximate Global Size

Figure 23 – Corner node S11 stress values for varying mesh densities.


17

Figure 25 - Corner node S12 stress values for varying mesh densities.

3.4

3.42

3.44

3.46

3.48

3.5

3.52

3.54

3.56

3.58

80 40 20 15 10 9

Stre

ss (

N/m

m2)


2.85

2.9

2.95

3

3.05

3.1

3.15

3.2

3.25

80 40 20 15 10 9

Stre

ss (

N/m

m2)


Figure 24 - Corner node S22 stress values for varying mesh densities.

For the corner node, it can be seen that all the stress values were essentially converged (to within

2%) at the approximate global size of 20. This was assuming that the optimum reference value was

given by a global size of 9, as no finer mesh was allowable in this situation.

For the left-hand cap-base node, the S11 and S22 elements were also within 2% of optimum at the

global size of 20. The convergence of the S12 element, however, was quite far out; it can be seen

that the value changed by a large amount between mesh iterations, indicating a lack of convergence.

For the right-hand cap-base node, the S11 values were within 4% of optimum at a global size of 20,

but the S22 values were only within 14% and the S12 values were within 9%. Again, these values

varied significantly between each mesh iteration, so using a finer mesh just to satisfy these values

would likely be a waste of computer resources.

The next step from here would be to refine the mesh around the areas of maximum stress, such as

the lower end of the end caps and around the pipe connection holes. Some values from these areas,


18

as mentioned above, can be seen as changing significantly even between the very fine meshes from

a global size of 9 and 10, so these areas may benefit from a finer mesh. This is studied in section 5.4.

5.3. Design Iterations

First iteration: The initial Excel optimised layer thickness of 1.22mm gave a minimum spreadsheet

FoS (Factor of Safety) of 1.205 – this was as close to 1.2 as possible without going in to unrealistic

decimal places for the thickness of each layer. This layer thickness provided an Abaqus model that

only output stresses equating to a FoS of 0.124. Hence, the Excel thickness needed increasing

significantly to give an Abaqus FoS of 1.2. (Refer to Excel sheet “AbaqusFoS(1)”)

Second iteration: To understand the relationship between the Excel and Abaqus FoS outputs, the

layer thickness was multiplied by 8 to get 9.76mm, as this figure was about as thick as was possible

to go without losing the characteristic of the pressure vessel being thin-wall. This thickness gave a

spreadsheet FoS of 9.643, which provided an Abaqus FoS of 1.249. (Refer to Excel sheet

“AbaqusFos(2)”)

Third iteration: The values from the second iteration were compared with the figures from the first

iteration, and a relationship was drawn up between them via a set of equations.

Firstly, the multiplication factors of each iteration’s respective FoS values were calculated.

These multiplication factors determined by how much the Abaqus FoS (output) needed to be

multiplied by to equal the Excel FoS (input).

Next, as the multiplication factors were different, the rate of change of this factor was

determined, for every 1 ‘increment’ of FoS value (note: FoS does not have units, it is

essentially a ratio). It is interesting to note here that the multiplication factor decreases with

an increase in layer thickness.

From this rate of change value, the multiplication factor for an Abaqus FoS of 1.2 was

calculated.

This multiplication factor was then used to determine the desired FoS within Excel to

achieve an FoS of 1.2 output from Abaqus.

Please refer to Excel sheet “AbaqusFoS(3)” for these calculations and the following data.

This set of equations led to an optimum Excel FoS of 9.362, to be able to achieve an Abaqus FoS of as

close to 1.2 as possible. The smallest thickness (to within 2 decimal places) that could provide this

FoS was a thickness of 9.48mm, which provided an Excel FoS of 9.367. A thickness of 9.47mm gave

an Excel FoS of 9.357, which was too low.

Inputting this layer thickness of 9.48mm in to Abaqus resulted in a minimum Abaqus FoS of 1.210,

hence the design was now fully optimised for this stage of analysis.

5.4. Altering the Concrete Support Positions

All Abaqus modelling up to this point had been done under the assumption of the vessel being

contained by two radial concrete support structures mounted 1600mm apart (hence each one was

800mm from the centre of the vessel). An investigation was undertaken to alter these concrete

support positions and observe the effect of these alterations on the FoS values output by the Abaqus


19

simulation. The concrete supports were changed to be mounted 1500mm and 1400mm apart (hence

each 750mm and 700mm, respectively, from the centre of the vessel). These alterations were done

to the model by changing the position of the face partition that was created to simulate the concrete

supports. The pressurisation simulations were run on both additional models (named “D1500” and

“D1400” with respective job names of “3-D1500” and “4-D1400”) and the stress results compared to

the original D1600 model. These 3 models are shown below in fig.26. One can observe the radial

concrete boundary constraints moving slightly inwards on each consecutive model.

Figure 26 - Screenshot images of each deformed vessel with differing concrete support boundary conditions. D1600 is top, D1500 middle, and D1400 on bottom. Contour plots are showing S11 element for ply 8.


20

When comparing the D1500 model to the D1600 (data shown in sheet named AbaqusFoS(D1500) in

the Excel spreadsheet), the FoS was found to be almost identical; it had dropped by only 0.008, from

1.210 to 1.202. Hence, changing the concrete supports to be 1500mm apart made no real difference

to the model developed by the author.

However, when comparing the D1400 model to the D1600 (data shown in sheet named

AbaqusFoS(D1400) in the Excel spreadsheet), the FoS had improved quite significantly, to

approximately 1.41. This was primarily down to an approximate 20% reduction in the S11 stress

element for ply 8 (the layer that would fail first). However, the author believes this to be an anomoly

with the way Abaqus models this area of the vessel, as discussed below.

The reduction of stress, in this case, was caused by the insufficiently refined mesh around the centre

hole and edges, that can be seen in the D1400 vessel at the bottom of fig.26. However, through

some experimentation with refining the mesh around the centre hole, it was found that all stress

values here kept increasing as the density of the mesh increased. The following ply 8 S11 stresses

were noted down with an increase in mesh density each time: 6.886, 8.013, 8.890, 9.546, 10.22. This

suggests the stress values will continue to increase indefinitely, indicating a point of infinite stress

concentration somewhere around the hole. Fig.27 below shows this area refined with a very fine

local mesh size of approximately 5, which gave the ply 8 S11 value of 9.546.

Overall, however, the range of stress values for the D1400 model are roughly equal to the other two

models. It is difficult to draw absolute conclusions from these tests, as the way in which Abaqus

models each vessel is subtly different, so the results aren’t directly comparable. In the author’s

opinion, however, changing the concrete support positions had negligible effect on the FoS of the

pressure vessel, as exhibited by the similarity between the D1500 and D1600 models and the

average FoS values of all the models.

Figure 27 - Very fine mesh around centre hole on D1400 model - an area of potentially infinite stress concentration.


21

6. Connecting Pipework to Inlets and Outlets

Pressure vessels must include some pipe fittings to allow the intended contents of the vessel to flow

in and out; however, a cylindrical pressure vessel is much weaker when it has holes in it. The

maximum stress wil be much larger if there is a circular hole in the shell, causing a rise in the stress

distribution around the hole. (A. Kharat, 2013)

In metallic pressure vessels, metallic pipe bosses are usually welded or threaded in to position.

However, a composite material cannot be welded, so the usual method of boss attachment is to

either chemically bond the boss in place, or utilise some form of attachment flange. They can also be

integrally wound in to the vessel by designing the layup pattern to encase the bosses within the

vessel wall during the filament winding process, or by using a metallic inner liner pre-formed with

integral bosses, such as that shown in fig.28 below.

The pipe connection holes in the author’s Abaqus model have been modelled with ‘Encastre’ type

boundary conditions, to fix them in place in all directions of potential deformation. This is due to the

assumption that real vessel pipes will be fixed in place via supports and solid input/output bosses,

and would not be designed to move.

Figure 28 - Carbon fibre filament wound pressure vessel (cutaway) with metallic liner featuring integral pipe bosses (Composites World, 2012)


22

7. Considering the Weight of the Vessel

For more accurate results, the weight of the vessel was considered in the Abaqus simulation. This

was done by adding a ‘Gravity’ load, by going to the Load module and selecting ‘Create Load’, then

choosing ‘gravity’ for the ‘load type’. In the ‘Edit Load’ dialog box, component 3 (the Z-axis) was

given a value of 9.81 to simulate the gravitational effect of Earth’s mass on the vessel. Components 1

and 2 were left at 0, as gravity only acts in the axis perpendicular to ground. Gravity was applied in

the Z-axis as this is the axis parallel to that of the holes in the top and bottom of the vessel, as shown

in the design brief diagrams in fig.6 on page 7, correctly representing its orientation in the real

world.

An issue arises when considering gravity, however, due to the way the pressure vessel was originally

modelled as an eighth of the total vessel. The effect of gravity means that only half of the model

(split along the longitudinal XY plane) is actually being accurately modelled. Due to the symmetry

boundary conditions, gravity is effectively acting towards, or from, the central XY plane of the vessel,

NOT to or from the virtual ground. If the gravity values are vastly exaggerated to 9810 and -9810,

and the mirror planes are activated on the deformed model view, one can see this problem visually

when comparing the effect of plus and minus gravities, as shown in fig.29 below.

The effect shown in fig.29 is not observable when the gravity values are set correctly at -9.81 and

+9.81 because they have such a small effect on the overall stresses/deformation. The negative

gravity can be seen as pulling the vessel wall inwards, as the gravity is effectively acting inwards,

towards the centre XY plane of the vessel. The positive gravity can be seen as pushing the vessel

outwards, effectively meaning gravity is acting outwards from the centre plane of the vessel.

Depending on whether a minus or positive gravity value is set in the gravity load, either the top half

only or the bottom half only of the vessel is being effectively modelled. To get round this issue,

analysis needs to be done with both a negative and positive gravity value, to see the effect of gravity

on both upper and lower halves of the pressure vessel. An image has been created below in fig.30

that represents, roughly, what the model would look like if the whole vessel, or at least a quarter

Figure 29 – Side view of pressure vessels showing effect of negative (top) and positive (bottom) gravity. Negative gravity models the top half of the vessel correctly, whereas positive gravity models the bottom half of the vessel correctly.


23

Figure 30 - The positive/negative gravity vessels superimposed on to each other to create representative image of (exaggerated) realistic deformation

model reflected in the XZ and YZ planes, would have been modelled instead of an eighth model

(image still shows exaggerated deformation due to excessive gravity).

The stress data from the two Abaqus models are shown in Excel sheets “AbaqusFoS(-Gravity)” and

“AbaqusFoS(+Gravity)”.

The effect of both positive and negative gravity had negligible effect on the stresses output by

Abaqus. This is likely due to the fact that the force exerted by gravity on the vessel is very small in

comparison to the forces exerted by the internal pressure on the vessel. Nonetheless, the FoS values

did change very slightly, with the negative gravity (top half of vessel) giving a more favourable FoS of

1.21070 (when compared to no gravity giving 1.21018), and the positive gravity (bottom half of

vessel) giving a less favourable FoS of 1.20967.

These results agree with the theoretical effect of gravity on each half of the vessel. The top half of

the vessel would effectively have gravity helping to ‘cancel out’ the pressure components acting

upwards on the inside of this half of the vessel. However, the bottom half of the vessel would

effectively have gravity helping to ‘reinforce’ the pressure components acting downwards on the

inside of this half of the vessel. This means that realistically modelled gravity would:

Reduce wall stresses, increasing the FoS, in the upper half of the vessel,

Increase wall stresses, decreasing the FoS, in the lower half of the vessel.

The effect of gravity acts (essentially) equally on each vessel half, as the FoS was increased by

0.00052 in the top half, and decreased by 0.00051 in the lower half, meaning that having no gravity

effect actually provides a perfectly reliable average FoS for the entire vessel.


24

8. Consideration of Environmental Temperature Change

The first step in the consideration of temperature change within the Abaqus model was to assign the

coefficients of expansion to the composite material. This was done as follows:

Firstly, in the ‘Edit Material’ dialog box for the “Composite” material,

‘Mechanical>Expansion’ was selected from the drop down list for material behaviours.

In the ‘Expansion’ section, ‘Orthotropic’ was selected from the ‘Type’ drop down list.

Alpha11 and Alpha22 values (-4e-6 and 57e-6, respectively, as specified in the project brief)

were then input. Alpha33 was set to 0.

OK was then clicked to save these material property changes (shown below in fig.31).

Now that the pressure vessel material had the relevant expansion coefficients, the ‘Initial’ and

‘Pressurise’ steps could be modified to include a temperature change.

Firstly, within the Initial step, in ‘Predefined Fields’, a new field was created, called

‘Start_Temp’.

In the ‘Create Predefined Field’ dialog box, the category of ‘Other’ was chosen, then

‘Temperature’ was selected as the type for selected step.

The whole model was then highlighted to indicate the regions on which the temperature

change would occur.

In the ‘Edit Predefined Field’ dialog box, the drop-down settings were left at their defaults,

and a magnitude of 0 was input.

OK was then clicked to save this predefined field (shown below in fig.32).

Figure 31 - Screenshots of expansion coefficient additions to composite material properties


25

This first step was to create a reference temperature of 0. The actual temperature value doesn’t

matter here, only the difference in the values input in to the initial step and the pressurise step. The

procedure above was, therefore, also performed in the ‘Pressurise’ step, creating a predefined field

called ‘Finish_Temp’, which was set to a magnitude of 20. This was to simulate a temperature

increase of 20 degrees during the pressurise step. A temperature decrease was also subsequently

modelled, by simply changing the initial start value to 20 and the finish value to 0.

The stress data from these two temperature change Abaqus models are shown in Excel sheets

‘AbaqusFoS(TempIncrease)’ and ‘AbaqusFoS(TempDecrease)’.

For the temperature increase model, the Abaqus simulation output all S22 stress components in

negative values, indicating that they were all in compression. This caused errors in two of the Tsai-

Wu FoS calculations in Excel, due to negative square root values (which are not possible to

calculate). However, from the Stress FoS values calculated for these two layers (2 and 7), it could be

seen that they were highly unlikely to be the cause of failure within the laminate. The layer that has

been the cause of failure for all previous models has been Ply 8; this was the ply that again gave the

lowest FoS in this simulation, with a value of 1.155. Henceforth, these Tsai-Wu errors are of no great

importance.

The Abaqus FoS output for the 20 degree temperature increase model correlated well with the Excel

FoS output for a 20 degree temperature increase model. When compared to having no temperature

effect, the Excel model FoS decreased from 9.367 to 6.767, and the Abaqus model FoS also

decreased from 1.210 to 1.155.

The Abaqus FoS output for the 20 degree temperature decrease model also correlated well with the

Excel FoS output for a 20 degree temperature decrease model. When compared to having no

temperature effect, the Excel model FoS increased from 9.367 to 15.380, and the Abaqus model FoS

also increased from 1.210 to 1.290.

Figure 32- Screenshot of Predefined Field settings to represent initial temperature reference.


26

Although the Abaqus and Excel models correlate well in terms of increasing/decreasing FoS, the

Excel model does change by a much larger percentage than the Abaqus model does. These results

do, however, show evidence that the Abaqus model follows the theoretical behaviour simulated in

the original Excel spreadsheet model.

It was assumed for the temperature effect investigation that the temperature change had negligible

effect on the pressure of the substance being stored within the pressure vessel.

The results from this temperature effect investigation show that composite filament wound pressure

vessels should always be designed with reference to the hottest temperature they are expected to

withstand (relative to average ambient operating temperature), as any temperature below this will

improve the effective factor of safety of the vessel. Whether that maximum temperature is in the

sunshine on a hot summer’s day, or in the inferno of a raging fire, is dependent on the vessels

application and the specifications set by the original design team.


27

9. Conclusions and Discussion

The studies detailed in this report have shown several points of interest.

It can be seen that the Excel spreadsheet model and the Abaqus FEA model correlate well on most

areas of experimentation. However, the modelling of the holes on the Abaqus models greatly

increases the stresses output by the FEA simulation, which is due to the fact that holes in a pressure

vessel cause areas of high stress concentration, creating points of mathematically infinite stress. This

phenomenon has been witnessed in the effects of varying mesh density around the centre holes of

the vessel and finding that the stresses continue to increase with increasing mesh density. However,

holes are an absolute necessity to include in a pressure vessel, due to the obvious issue of needing

to get the contents of the vessel in and out. The holes in the end caps, however, pose much less of a

problem than those positioned in the centre cylindrical section of the vessel, so these types of hole

placements should be avoided in pressure vessel designs.

The study in to altering concrete support positions found that the position of the concrete supports

does not greatly affect the stresses imparted in to the vessel, at least in relation to the changes

made during this study.

It has been found that the effects of gravity make very little difference to the consideration of

stresses in the vessel walls. This is due to the fact that the resultant forces exerted on the vessel by

gravity are much less than the resultant forces exerted on the vessel by the internal pressure.

The effects of temperature increasing and decreasing have been found to have a considerable effect

on the stresses within the vessel walls. This led to the conclusion that the vessel should be designed

with reference to the maximum temperature at which its designers wish it to be able to withstand.

The effect of how Abaqus models curved faces will also create inaccuracies in the results provided by

Abaqus, as any computer software can only model a curved face as a series of small straight lines.

Contrast this to the Excel simulation, which used mathematics based upon perfectly smooth curved

surfaces.


28

Bibliography

A. Kharat, V.V.K. (2013) Stress Concentration at Openings in Pressure Vessels - A Review.

International Journal of Innovative Research in Science, Engineering and Technology, 2(3), p.670.

Abaqus. (unknown) 4.4 Mesh Convergence [Online]. Available from:

http://www.maths.cam.ac.uk/computing/software/abaqus_docs/docs/v6.12/books/gsk/default.htm

?startat=ch04s04.html [Accessed 01 December 2014].

Amali, R. (2014) W13 Finite Element Analysis of Composite Materials. UFMFWD-30-M Modelling and

Simulation [Online]. Available from: https://my.uwe.ac.uk/ [Accessed 21 November 2014].

Composites World. (2012) 2012 JEC Europe Highlights [Online]. Available from:

http://d2n4wb9orp1vta.cloudfront.net/resources/images/125/cdn/cms/0512HPC_JEC_pressure_ves

sel.jpg [Accessed 06 December 2014].

eFunda. (2014) Thin-Walled Pressure Vessels [Online]. Available from:

http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/pressure_vessel.cfm [Accessed

22 November 2014].

Ferdinand P. Beer, E.R.J.J..J.T.D.D.F.M. (2012) Hooke's Law; Modulus of Elasticity. In Mechanics of

Materials. 6th ed. New York: McGraw-Hill. pp.62-64.

M.Madhavi, K.V.J.R.K.N.R. (2009) Design and Analysis of Filament Wound Composite Pressure Vessel

with Integrated-end Domes. Defence Science Journal, 59(1), pp.73-81.

McClean Anderson. (2014) What is Filament Winding? [Online]. Available from:

http://mccleananderson.com/index.cfm?pid=31&pageTitle=What-is-Filament-Winding? [Accessed

21 November 2014].

Nuplex Industries Ltd. (2014) Filament Winding [Online]. Available from:

http://www.nuplex.com/Composites/processes/filament-winding [Accessed 22 November 2014].

unknown. (unknown) Stiffness Reduction of the Laminates [Online]. Available from:

http://www.pmi.lv/soft/stirel/ [Accessed 22 November 2014].

http://www.maths.cam.ac.uk/computing/software/abaqus_docs/docs/v6.12/books/gsk/default.htm?startat=ch04s04.html

http://www.maths.cam.ac.uk/computing/software/abaqus_docs/docs/v6.12/books/gsk/default.htm?startat=ch04s04.html

https://my.uwe.ac.uk/

http://d2n4wb9orp1vta.cloudfront.net/resources/images/125/cdn/cms/0512HPC_JEC_pressure_vessel.jpg

http://d2n4wb9orp1vta.cloudfront.net/resources/images/125/cdn/cms/0512HPC_JEC_pressure_vessel.jpg

http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/pressure_vessel.cfm

http://mccleananderson.com/index.cfm?pid=31&pageTitle=What-is-Filament-Winding?

http://www.nuplex.com/Composites/processes/filament-winding

http://www.pmi.lv/soft/stirel/


29

Appendices

Appendix A

35

35.5

36

36.5

37

37.5

38

38.5

39

80 40 20 15 10 9

Stre

ss (

N/m

m2 )


Figure A1 - Mesh convergence graph for S11 stress element for left-hand end-cap baseline node.

3.42

3.44

3.46

3.48

3.5

3.52

3.54

3.56

3.58

3.6

3.62

80 40 20 15 10 9

Stre

ss (

N/m

m2 )



0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

80 40 20 15 10 9

Stre

ss (

N/m

m2 )




30

36

37

38

39

40

41

42

80 40 20 15 10 9

Stre

ss (

N/m

m2 )


Figure A4 – Mesh convergence graph for S11 stress element for right-hand end-cap baseline node.

0

2

4

6

8

10

12

80 40 20 15 10 9

Stre

ss (

N/m

m2)


Figure A5 - Mesh convergence graph for S22 stress element for right-hand end-cap baseline node.

-9.5

-9

-8.5

-8

-7.5

-7

80 40 20 15 10 9

Stre

ss (

N/m

m2)


Figure A6 - Mesh convergence graph for S12 stress element for right-hand end-cap baseline node.

FEA Coursework Report (Sven Cumner 11011794)

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