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UNIVERSITY OF CAPE TOWN

Research Project (CIV4044S)

Analysis of Stress Distribution in Parabolic and

Ellipsoidal Concrete Domes.

Undergraduate Thesis in the Faculty of Engineering and the Built

Environment

Prepared for: Department of Civil Engineering

Special field: Shell Structures

Supervisor: Prof. A. ZingoniPrepared by: Nazeer Slarmie

Student number: SLRMOG001

Date: 12th

November 2012

Thesis in partial fulfilment of the requirements for the degree ofBSc (Eng) Civil Engineering



I

Plagiarism Declaration

1. I know that plagiarism is wrong. Plagiarism is to use another‟s work and pretend that

it is one‟s own. 2. I have used the Harvard convention for citation and referencing. Each contribution to,

and quotation in, this dissertation from the work(s) of other people has been

attributed, and cited and referenced.

3. This dissertation is my own work.

4. I have not allowed, and will not allow, anyone to copy my work with the intention of

passing it off as his or her own work.

5. I acknowledge that copying someone else‟s work or part of it, is wrong, and declare

that this is my own work.

Course: CIV4044S (Research Project)

Supervisor Name: Prof. A Zingoni

Student Name: Nazeer Slarmie

Student number: SLRMOG001

Date: 12th

November 2012

Signed by: M.N. Slarmie

Signature:



II

Abstract

Concrete domes have been used as roofing solutions for many thousands of years. The thin

concrete shell is well suited for roofing due to its high strength-to-weight ratio and provides a

reduction in material used falling in line with the current trend of environmental

sustainability and provides an aesthetically pleasing finish to any structure. In this

dissertation, the stress distribution for the parabolic and ellipsoidal dome was analysed. This

was achieved by defining the hoop and meridonal stress resultants for the paraboloid in terms

of a single governing geometric property. Non-dimensional stress charts for values of lambda

covering ranging from zero to three were plotted and trends with in these charts were

discussed. Design tables for were generated to supply those who need to determine the stress

at any point along the dome profile. However in the case of of the ellipsoidal dome, it was not possible to express the equations for hoop and meridonal stress in terms of a governing

parameter. All attempts are illustrated in this dissertation.



III

Acknowledgements

This thesis represents the final requirement in obtaining my civil engineering degree and is

the culmination of 4 my four years as an undergraduate at the University of Cape Town. It

was done under the guidance of Professor Alphose Zingoni.

I would like to thank:

My supervisor

Professor Alphose Zingoni. For allowing me the opportunity to complete my thesis

under his supervision.

My close family

My mother: Shamiela Slarmie without her I wouldn‟t be where I am today. She

supported me throughout my undergraduate degree and provided me with more than I

could have asked for

My brother: Abdu-Raaziq Slarmie, when times were tough he was there for me and

now it is my turn to be there for him.

My grandfather: Abaabokir Smith, for raising me into who I am today. I am eternally

grateful.

My father: Osman Slarmie, for being there in any way he knew how.

My friends

Mahir Ebrahim, Shafeeq Mollagee, Naweed Kahaar and Taariq Solomons for making

the last 2 years exciting for me.

And last but not least, Sameeha Osman-Latib who got me out of so many tight spots



IV

Terms of Reference

On 7th of June 2012, a meeting was held with Professor Alphose Zingoni, initially it was

decided that an investigation on thickness variation within concrete domes will be conducted.

On 6th August 2012, a revised brief was obtained. The stress distributions in two different

shapes of concrete domes used as roofing solutions will be analysed and a parametric study

will completed in order to formulate design charts from which stresses in the domes may be

obtained. The proposed shapes for investigation were that of a parabolic dome of revolution

and an ellipsoidal dome of revolution.



V

List of Symbols

Symbol Description Unit

Radius of dome

Height of dome

Dome thickness

Surface area of dome

Weight of dome

Meridonal stress

Hoop stress

Tangential loading component

Radial loading component

Unit weight of concrete

Arc length

Ratio of dome height-to-radius

Meridonal radius of curvature

Hoop radius of curvature



VI

Glossary

Term Description

Meridian curve Generating curve of dome of revolution

Meridonal stress Stress developed in the tangential direction of

the shell

Hoop stress Stress developed in the radial direction

Paraboloid of revolution Surface obtained by rotating a parabola about

its axis

Ellipsoid of revolution Surface obtained by rotating an ellipse about

its axis



VII

Table of Contents

Plagiarism Declaration ............................................................................................................... I

Abstract ..................................................................................................................................... II

Acknowledgements .................................................................................................................. III

Terms of Reference .................................................................................................................. IV

List of Symbols ......................................................................................................................... V

Glossary ................................................................................................................................... VI

Table of Contents ................................................................................................................... VII

List of Figures ........................................................................................................................... X

List of Tables ........................................................................................................................... XI

List of Major Equations ........................................................................................................... XI

1 Introduction .................................................................................................................... 1-1

Description of investigation ............................................................................................... 1-2

2 Literature Review........................................................................................................... 2-1

2.1 Introduction to Shells .............................................................................................. 2-1

2.1.1 Classification of shells ..................................................................................... 2-1

Surfaces of revolution .................................................................................................... 2-2

Translational surfaces .................................................................................................... 2-3

Ruled surfaces ................................................................................................................ 2-3

2.1.2 Qualitative description of shell behaviour ....................................................... 2-3

2.2 Shell Theory ............................................................................................................ 2-4

2.2.1 Linear shell theory ........................................................................................... 2-5

2.2.2 Membrane Theory ............................................................................................ 2-6

2.2.3 Bending Theory ............................................................................................... 2-7

2.3 Membrane analysis of shells of revolution under axisymmetric loading................ 2-8

2.3.1 Geometric description of shells ....................................................................... 2-8

2.3.2 Governing membrane equations ...................................................................... 2-8



VIII

2.3.3 Deformation of Shells ...................................................................................... 2-9

2.4 Membrane analysis of concrete domes of revolution.............................................. 2-9

2.4.1 Spherical domes ............................................................................................. 2-10

3 Development of closed form solutions .......................................................................... 3-1

3.1 Assumptions ............................................................................................................ 3-1

3.2 Geometric Properties ............................................................................................... 3-1

3.2.1 Curvature.......................................................................................................... 3-1

3.2.2 Surface area ...................................................................................................... 3-1

3.2.3 Loading ............................................................................................................ 3-3

3.3 Method of sections for determination of stresses .................................................... 3-3

3.4 Parameterisation of stresses equations .................................................................... 3-5

4 Parabolic Dome of Revolution....................................................................................... 4-1

5 Ellipsoidal Dome of Revolution .................................................................................... 5-1

5.1 Method of Sections.................................................................................................. 5-1

5.2 The general stress equations for membrane theory ................................................. 5-3

6 Parametric Results ......................................................................................................... 6-1

6.1 Parabolic Dome of Revolution ................................................................................ 6-1

6.2 Ellipsoid of revolution ............................................................................................. 6-1

7 Discussion of Results ..................................................................................................... 7-1

7.1 Parabolic dome of revolution .................................................................................. 7-1

7.1.1 Meridonal stress ............................................................................................... 7-1

7.1.2 Hoop stress ....................................................................................................... 7-5

8 Numerical Example ....................................................................................................... 8-1

9 Conclusion and Recommendations ................................................................................ 9-1

10 References .................................................................................................................... 10-1

Appendixes ................................................................................................................................. i

Appendix A: Design tables for parabolic dome ..................................................................... ii



IX

Meridonal stresses .............................................................................................................. ii

Hoop stresses ................................................................................................................. viii

Appendix B: Logbook............................................................................................................. xiv



X

List of Figures

Figure 1-1 Sheikh Zayed Mosque (Source: Nazeer Slarmie) ................................................ 1-1

Figure 2-1 Surfaces with positive, negative and zero Gaussian curvature (Source:Farshad) 2-1

Figure 2-2 Examples of developable and non-developable surfaces (Source:Farshad) ........ 2-2

Figure 2-3 Surface of Revolution .......................................................................................... 2-2

Figure 2-4 Examples of translational surfaces (Source: Farshad) ......................................... 2-3

Figure 2-5 Ruled surfaces ...................................................................................................... 2-3

Figure 2-6 Membrane and bending resultants on shell element(Source: Farshad) ................ 2-4

Figure 2-7 Geometrical and loading discontinuities (Source:Farshad) ................................. 2-6

Figure 2-8 Surface of revolution showing principal radii of curvature, meridian curves and

the parallel circles .................................................................................................................. 2-8

Figure 2-9 State of Internal force field under axisymmetric and non-axisymmetric loading(

Source: Farshad) .................................................................................................................. 2-10

Figure 3-1 Computation of surface area by segmentation (Source: Stewart) ........................ 3-2

Figure 3-2 Arc length element ............................................................................................... 3-2

Figure 3-3 Loading Components due to self-weight ............................................................. 3-3

Figure 3-4 Arbitrary shell profile ........................................................................................... 3-4

Figure 3-5 Force balance ....................................................................................................... 3-5

Figure 4-1 Parameters of defined parabolic meridian ............................................................ 4-1

Figure 5-1 Parameters of defined ellipsoidal meridian .......................................................... 5-1

Figure 5-2 Output of online integral calculator (Source: Wolfram Alpha) ........................... 5-4

Figure 7-1 Meridonal stress distribution in parabolic dome .................................................. 7-3

Figure 7-2 Peak meridonal stress variation with lambda ....................................................... 7-4

Figure 7-3 Hoop stress distribution in parabolic dome .......................................................... 7-7

Figure 7-4 Peak hoop stress variation with lambda ............................................................... 7-8



XI

List of Tables

Table 1 Range of phi values defining parabolic dome .......................................................... 7-1

Table 2 Peak meridonal stress for corresponding values of lambda ...................................... 7-2

Table 3 Peak hoop stress for corresponding values of lambda .............................................. 7-5

List of Major Equations

Equation (2.1) General meridonal stress equation ................................................................ 2-9

Equation (2.1) General hoop stress equation ........................................................................ 2-9

Eqaution (3.9) Meridonal stress equilibrium equation ......................................................... 3-5

Equation (4.11) Meridonal stress equation for parabolic dome ............................................ 4-3

Equation (4.12) Hoop stress equation for parabolic dome ................................................... 4-4

Equation (5.5) Incomplete equilibrium equation for ellipsoidal dome ................................. 5-2

Equation (5.9) Invalid meridonal stress equation for ellipsoidal dome ................................ 5-4



1-1

1 Introduction

Shell structures are used in many engineering disciplines; however, considering specifically

the civil engineering field, concrete domes have been used as roofing solutions for many

thousands of years. The oldest dome still in existence, The Pantheon in Rome, Italy; has been

standing for two thousand years and is still in use today. Whether it is for religious roofing,

housing or any other purpose, the thin concrete shell is well suited due to its high strength-to-

weight ratio and provides a reduction in material used falling in line with the current trend of

environmental sustainability. The thin shell is not constrained to the hemispherical dome; it

can be manipulated into various arbitrary shapes such as that seen at the Sydney Opera House

in Australia. The hemispherical dome is in essence the simplest type of dome and provides an

aesthetically pleasing finish to structures as at the Sheikh Zayed Grand Mosque, Abu Dhabi.

Figure 1-1 Sheikh Zayed Mosque (Source: Nazeer Slarmie)

The benefits of concrete domes are many, however, replicating nature‟s elegance while

maintaining the structural properties required for the elaborate designs required in the 21st

century has been challenging. This is mainly due to the fact that designing of concrete domes

can become cumbersome due to the complicated underlying theory and whether or not the

theory is valid for the design in question. Advancements in technology have allowed for the

numerical analysis of complicated designs with the aid of Finite Element Analysis. Computer programmes have been coded to use the finite method to solve practical engineering



1-2

problems; however the problem arises when inexperienced designers make use of software

and obtain results but are either incorrectly inputting the conditions or misinterpreting the

results. It thus follows that even though modern day computer programmes have the potential

to solve many problems, a fundamental understanding of the underlying theory is required to

make judgment calls on the defining properties of the shell as well as on the interpretation of

the results.

Description of investigation

The topic under investigation is that of stress variation in the concrete paraboloid and

ellipsoid of revolution for use as roofing solutions for residential, commercial or industrial

use. Once the stress equations for each shape are analytically obtained, the stress distribution

along the profile of the dome will be plotted so that a designer may read the stress values off

the charts and design the structure accordingly. This investigation will focus only the forces

developed in the membrane of the shell which are primarily generated due to the self-weight

of the structure. Due to constraints in time as well as knowledge, the effects of wind loading,

which induce bending forces within the structure, will be ignored.

The ultimate objective is the creation of design charts and the provision of recommendations

to aid the engineer looking to design a structure of the geometric shape in question.

Figure 1-2 Parabolic dome used as housing



2-1

2 Literature Review

2.1 Introduction to Shells

“A shell may be defined as a three-dimensional structure bounded primarily by two arbitrary

curved surfaces a relatively small distance apart.” (Zingoni, 1997, p. 10)

Shell structures support external loads by using their geometrical form to transfer loads to the

supports, it is for this reason they are called form resistant structures. (Farshad, 1992, p. 15)

The geometrical properties that differentiate shells from other structural forms such as plates

and beams are the possession of surface and curvature. Curvature allows for load

transmission by in-plane action which limits flexural action, and also gives rise to beneficialmechanical properties such as a high strength to weight ratio and rigidity.

2.1.1 Classification of shells

Shell surfaces are classified using the definition of Gaussian curvature. The curvature of the

shell can be quantified by the equation

(2.1)

The numerical value is not of real significance however whether the curvature is positive

negative or zero is what is used in the classification. Surfaces with positive, negative and zero

curvature are respectively called a synclastic surface, anticlastic surface or a zero Gaussian

surface. This can be seen in figure 2-1 below.

Figure 2-1 Surfaces with positive, negative and zero Gaussian curvature (Source:Farshad)

In addition to this concept, surfaces may also be classified based on their geometrical

developability. Surfaces can either be developable or non-developable. (Farshad, 1992) states

that a developable surface is one which can be developed into a plane with cutting or



2-2

stretching the middle surface where as a non-developable surface has to be cut or stretched to

achieve the planar form.

Figure 2-2 Examples of developable and non-developable surfaces (Source:Farshad)

Surfaces with double curvature are usually non-developable whereas surfaces with single

curvature are always developable.

From a structural point of view non-developable surfaces require more external energy to

deform compared to a developable shell. Based on this, one can conclude that non-

developable shells are generally stronger and more stable than a developable shell having the

same over all dimensions. (Farshad, 1992)

The final classifications for shells are into surfaces of revolution, translational surfaces and

ruled surfaces.

Surfaces of revolution

Theses surfaces are generated by a plane curve, known as the meridonal curve which is

rotated about an axis called the axis of revolution.

Figure 2-3 Surface of Revolution



2-3

Translational surfaces

„These surfaces are generated by sliding a plane curve along another plane curve while

keeping the orientation of the sliding curve constant.‟ (Farshad, 1992) In the case where the

curve where the surface is slid along is a straight line, it may be called a cylindrical surface.

Figure 2-4 Examples of translational surfaces (Source: Farshad)

Ruled surfaces

These surfaces are obtained by sliding a straight line, whose two ends remain on two

generating curves, in such a way that it remains parallel to a chosen direction.

Figure 2-5 Ruled surfaces

2.1.2 Qualitative description of shell behaviour

The load carrying mechanism of a shell can be split into 2 groups, namely the internal

bending force field and the internal membrane force field. Each has components as follows:

Bending Field: Mx, My, Mxy, Myx, Qx, Qy



2-4

Membrane Field: Nx, Ny, Nxy, Nyx

This is depicted on a shell element below.

Figure 2-6 Membrane and bending resultants on shell element(Source: Farshad)

For any object in space there are six governing equilibrium equations. Since there are more

than six force resultants, it is safe to say that shells, in general, are internally statically

indeterminate structures. According to (Farshad, 1992) although internal force redundancy is

an indication that there are additional load carrying mechanisms present, it is not needed to

achieve equilibrium in the shell.

2.2 Shell Theory

There are many established shell theories which attempt to analyse structural shell behaviour.

According to (Farshad, 1992) the factors which have influences in these shell theories are

material type and behaviour

shell geometry

Loading conditions

Deformation ranges

Desired shell behaviour

Computational means

(Farshad, 1992) also says that any shell theory is founded on three sets of relations; namely,

the equilibrium equations, kinematical relations and constitutive relations, these relations

along with boundary conditions form the completed shell theory.

“Most common shell theories are those based on linear elasticity concepts. Linear shell

theories adequately predict stresses and deformations for shells exhibiting small elastic

deformations, that is, deformations for which it is assumed that the equilibrium-equation

conditions for deformed elements are the same as if they were not deformed and Hooke‟s-law

applies.” (Baker, Kovalevsky, & Rish, 1972) These linear concepts are more prevalently used



2-5

than the non-linear counterparts due to the fact that they are easier to solve which makes it

more feasible to use.

According to (Baker, Kovalevsky, & Rish, 1972) when approaching a shell problem, the

development of exact theoretical expressions does not always help with practical shell

problems due to the fact that these equations cannot always be solved and if they can, it might

only be valid for special cases. The same applies to experimental data, which also cannot be

done for all cases. Difficulties in theory and experiment have led to applied engineering

methods for the analysis of shells. “Although these methods are approximate, and only valid

under specific conditions, they generally are very useful and give good accuracy for the

analysis of practical engineering shell structures.” (Baker, Kovalevsky, & Rish, 1972)

2.2.1 Linear shell theory

“The theory of small deflections of thin elastic shells is based upon the equations of the

mathematical theory of linear elasticity.” (Baker, Kovalevsky, & Rish, 1972) Due to the

geometry of shells, the three-dimensional elasticity equations need not be considered (using

them leads to complicated equations which cannot be easily adapted to practical problems).

Simplification of the problem is accomplished by reducing the shell problem to the study ofthe middle surface of the shell. The starting point is always the general three-dimensional

equations of elasticity; this then gets simplified by reducing the general system of equations

containing three space variables to that of only two space variables.

Below are the classic assumptions for the first-order approximation shell theory.

A.E.H Love was the first investigator to present a successful approximation shell theory

based on classical elasticity. To simplify the constitutive relations he proposed assumptions

which are commonly referred to as the Kirchhoff-Love hypothesis. The assumptions are as

follows:

1. Shell thickness t is negligibly small in comparison to the least radius of curvature of

the middle surface.

2. Linear elements normal to the unstrained middle surface remain straight during

deformation and suffer no extensions.

3. Normals to the undeformed middle surface remain normal to the deformed middlesurface.



2-6

4. The component of stress normal to the middle surface is small compared with other

components of stress and may be neglected in the stress-strain relationships.

5. Strains and displacements are small so that quantities containing second and higher-

order terms are neglected in comparison with first-order terms in the strain equations.

2.2.2 Membrane Theory

For any object in space, there are 6 governing equilibrium equations, and there are more than

6 resultants so generally, a shell is internally statically indeterminate. (Farshad, 1992)

According to (Farshad, 1992, p. 16), if one considers a loading case which only induces the

membrane field, three of the equilibrium equations is satisfied (i.e. all moments equal zero).

This leads to only 3 resultants in the membrane field, namely: Nx, Ny and Nxy=Nyx. This

causes the internal system to be statically determinate and allows determination of forces by

use of only the equilibrium equations. This is in essence Membrane Theory

For the membrane theory to be valid, certain loading, boundary and geometrical conditions

need to be satisfied. According to (Farshad, 1992) the most often violated conditions are:

Deformation constraints and boundary conditions incompatible

Application of concentrated forces and change in shell geometry or sudden changes in

curvature.

Figure 2-7 Geometrical and loading discontinuities (Source:Farshad)

In cases such as those depicted in figure 2-7 above, the membrane field of forces and

deformations would not be sufficient in to satisfy all equilibrium and displacement

requirements in the regions of equilibrium unconformity, geometrical incompatibility,

loading discontinuity and geometrical non-uniformity. Therefore the membrane theory will



2-7

not hold throughout such shells as these discontinuities and incompatibilities induce bending

components in to the shell which goes against the membrane theory.

(Zingoni, 1997, p. 25) Suggested a way to validate the membrane hypothesis in shells of

revolution under axisymmetric loading, he firstly calculates the meridonal and hoop stresses

for the membrane solution N N , using these values determines the meridonal

rotation (Vm), which is then used to determine the moments in the hoop and meridonal

directions M M which need to be negligible in comparison with extensional

stresses.

According to (Farshad, 1992) based on laboratory and field experiments as well as theoretical

calculations, the bending field developed at any of the above mentioned discontinuities are

localised around the area which the violation occurs and its effects weaken has one moves

further from the membrane non-conformity. The rest of the shell is virtually free from

bending and can be analysed as a membrane.

(Zingoni, 1997, p. 29) Also states that the support conditions required to conform to the

membrane theory is that only tangential force reactions are allowed at a supported edge.

If this is not the case, shear forces or moments may develop violating the theory.

2.2.3 Bending Theory

This theory is considered to be more general and exact than that of the membrane theory due

to the fact that all stresses are included in the analysis. The stresses include that of vertical

shear, bending and twisting. This leads to computational problems due to the complexity of

the equations; however simplifications can be made for rotationally symmetric geometries

subjected to rotationally symmetric loads. The following is an analogy on the shell-bending

action.

“A plate supported along the edges and loaded perpendicularly to the plate surfaces is

actually a two-dimensional equivalent of a beam supported at the ends and loaded

perpendicularly to the beam axis. In this case the plate, like the beam resists loads by two-

dimensional bending and shear. Beams resist loads by one-dimensional bending and shear.

The plate is a two-dimensional surface. A shell is also a surface but is three-dimensional.

Bending is resisted by the shell in a similar manner to the plate, except that for the plate,



2-8

bending is the main mechanism for resistance and for a shell it is only a secondary.” (Baker,

Kovalevsky, & Rish, 1972)

2.3 Membrane analysis of shells of revolution under axisymmetric loading

The types of shells encountered are typically double curvature domes which have positive

Gaussian curvature. They are used as roof coverage for sports walls, religious structures or

liquid containment structures. Pressure vessels may also be entirely constructed out of a shell

of revolution or have their end caps been made from a shell of revolution. Conical shells with

zero Gaussian curvature are used to cover liquid storage tanks as well as for the nose cones of

missiles.

2.3.1 Geometric description of shells

At any point on the shell of revolution, two principal radii of curvature can be defined

and. Where the curvature of the meridonal is curve and is the curvature of the parallel

circles.

Figure 2-8 Surface of revolution showing principal radii of curvature, meridian curves and the parallel circles

2.3.2 Governing membrane equations

Using the general equations for the theory of elasticity combined with the above geometric

properties, the general equations for the meridonal and hoop stresses can be quantified. It is

so defined that a positive hoop or meridonal stress indicates tension and a negative stress

indicates compression.

The general equations for a symmetrically loaded shell of revolution are developed by

(Ugural, 1981, pp. 203-204). These exact equations were also obtained by many authors

using the same principles



2-9

1 (2.1) General meridonal stress equation

N [∫ ( )]. (2.1)

2 (2.1) General hoop stress equation

(2.1)

This equation shows that the hoop stress is a function of meridonal stress, geometric

properties as well as the loading.

2.3.3 Deformation of ShellsThe equations for the displacement of a spherical dome under axisymmetric loading were

determined by using the hoop and meridonal stresses found above. These displacements were

initially in the hoop and meridonal direction, however that for the sake of practicality, the

equations were developed such that the horizontal shell displacement as well as the rotation

of the meridian (δ and V respectively) can be calculated.

( N

N

) (2.3)

* N – N

( N N )+ (2.4)

2.4 Membrane analysis of concrete domes of revolution

The membrane field of internal forces comprises of the meridonal, hoop and a membrane

shear force. However, under axisymmetric loading conditions, the membrane shear force iszero throughout the shell and the internal force field comprises of meridonal and hoop forces

only. The directions of principal normal stresses coincide with meridonal and hoop curve and

the shear stress will be zero along these directions.



2-10

Figure 2-9 State of Internal force field under axisymmetric and non-axisymmetric loading( Source: Farshad)

The structural behaviour of domes can be seen as the interaction of two mechanisms.

1. Arch action: This transfers loads from the top of the shell downwards along the

meridonal curve

2. Ring Action: This distributes the force along the circumference of the shell in the

hoop direction.

“The interaction of these two mechanisms gives rise to an efficient spatial behaviour of the

doubly curved shell.” (Farshad, 1992)

2.4.1 Spherical domes

(Zingoni, 1997, p. 97) Used the general stress distribution and displacement equations and

specified it to a dome, which resulted in the following equations:

2.4.1.1 Stresses under axisymmetric loading

Making use of the equations (x) and (x), Zingoni developed the stress distribution in a

spherical dome.

(2.9)

(2.10)



2-11

From the above equations, we can see that the meridonal stress ( N) is always implies the

entire dome is under compression in the meridonal direction. However the hoop stresses are

found to be in compression in the upper region of the shell and changes to tension at a certain

point ()2.4.1.2 Displacements under axisymmetric loading

(2.11)

(2.12)

2.4.1.3 Spherical Dome of gradually varying thickness

Zingoni analyses a spherical dome whose thickness varies along the length of its meridonal

curve. He considers that the variation of thickness only begins after a certain point along the

meridonal curve. This point can be considered to have 2-D polar co-ordinates (a,Φe )Where a

is the radius of the curve and Φe is the angle measured from the y-axis. The end point of the

curve of the curve is defined as (a, Φs).

In the section of the curve guided by co-ordinates (a, {0≤Φ≤Φe}), the weight of the shell is

considered to be uniform and the stresses due to self-weight can be calculated at any point

along this curve using Equations 2.5 & 2.6. In the remaining section of the curve guided by

polar co-ordinates (a, {Φs≤Φ≤Φe}), the thickness of the shell varies and is governed by the

equation

(2.13)

This variation in thickness causes the loading due to self-weight to vary in this section of theshell. This requires one to return to equations 2.1 and 2.2 and derive the equations for the

stress using this varying self-weight. (Zingoni, 1997, pp. 99-103)



3-1

3 Development of closed form solutions

3.1 Assumptions

Negative stresses imply compression and positive stresses imply tension.

Wind loading will not be taken into account

All bending effects will be ignored

No shear forces are developed due to the rotational symmetry of the domes

3.2 Geometric Properties

Before explicit solutions can be formed, the geometric properties of domes of revolution will

need to be quantified.

3.2.1 Curvature

The meridonal radius of curvature can be defined by the following equation

√

(3.1)

The hoop radius of curvature is defined by

(3.2)

3.2.2 Surface area

In order to compute the surface area of a shell of revolution, the shell can be segmented into

bands of which the length of the curve can be determined. This length is then multiplied the

circumference at the point in question. (Stewart, 2006)



3-2

Figure 3-1 Computation of surface area by segmentation (Source: Stewart)

An element of the curve can be isolated, if we assume its length equals ds, then by the

following relationship seen in the image below, ds may be written it terms of dx and dy.

Figure 3-2 Arc length element

√

The length of the curve can be found using the arc length formula

∫

(3.3)

The arc length is then multiplied by the circumference at the point giving the surface area of

the band, which when integrated over the domain of the curve leads to the surface area of

shell.



3-3

∫ √ (3.4)

3.2.3 Loading

The loading on a shell is solely due to the self-weight of the structure.

The self-weight of any concrete structure can be defined by the unit weight of the concrete (γ)

multiplied by volume of concrete. In a shell this volume is calculated by the surface area( S )

of the shell multiplied by shell thickness (t )

(3.5)

At any point on the surface of the shell, the weight is acting vertically downwards onto the

shell. This is the resultant force of and which acts in the tangential normal directionsrespectively. They can be quantified as

(3.6)

(3.7)

Figure 3-3 Loading Components due to self-weight

3.3 Method of sections for determination of stresses

In order to determine the stress distribution within a shell, the concept of equilibrium was

used to analyse the shape as described by (Zingoni, 1997) and by (Farshad, 1992). This

method is intuitive by nature as it relies simply on the balancing of forces within the shell. As



3-4

this research project only takes into account axi-symmetric loading, the only force the

structure will need to be analysed for is that of self-weight. This is purely based on membrane

theory as non axi-symmetric loading is not considered; however there are conditions to

whether the membrane theory holds under axi-symmetric loading as it is possible for

moments and shear forces to develop. These conditions were mentioned in earlier.

Consider a shell of revolution with an arbitrary meridian profile whose equation can be

defined by y= F(x) and with its axis of symmetry being the Y-axis. If a point T is identified

on this curve and a slice is taken at that point, i.e. the cap of the dome lying above point is

isolated.

Figure 3-4 Arbitrary shell profile

Based on Newton‟s Third Law, the force required by the shell below the slice to hold up the

cap is equal to force exerted by the cap on to the section below. From this we can say that

meridonal force at a point is generated due to the self-weight of the above cap. The self-

weight acts vertically while the meridonal force acts in a direction tangential to the meridian

curve at the point T .

Phi ( Φ ) can be defined as the angle between the positive axis of revolution and the normal to

the shell mid-surface at the point T as seen in Figure 3-4.



3-5

Figure 3-5 Force balance

The forces seen in Figure 3-5 are the self-weight of the cap (W ) measured in kN and the

meridonal force .which acts along the circumference of the shell and is measured in kN/m.

If we sum the forces in the vertical direction, we obtain the following equilibrium equation:

(3.8)

Therefore Ncan be written explicitly as

3 (3.9) Meridonal stress equilibrium equation

(3.9)

The stress N will need to be written explicitly in terms of Φ in order to see how stresses vary

along the profile of the curve.

3.4 Parameterisation of stresses equations

The stress equations found using the methods mentioned above will be written in terms of a

parameter which will be defined as the height of the shell over the radius of the base of the

shell. These stress equations will then be plotted against the angle (Φ) for each value of the

parameter within a practical range. The results will then be discussed.



4-1

4 Parabolic Dome of Revolution

The equation of the meridian curve for the parabolic dome can be defined by the equation

(4.1)

This equation needs to satisfy the following representation of the meridian curve where

: is the distance from the axis of revolution to curve at the base of the parabola, this

will further be referred to as the radius of the parabola.

: is the distance along the axis of revolution from the base of the parabola to its apex

and will further be referred to as the height of the parabola.

: is a place holding variable which will be rewritten in terms of a, b as these are the

variables we are mainly interested in

: Defined, as the angle from the positive axis revolution to the normal of the curve.

Figure 4-1 Parameters of defined parabolic meridian

The elimination of the place holding variable k can be done by evaluating the function at the

point on the base of the parabola (a, 0). Substituting these values into equation (3.3) leads to:

(4.2)

From which k can be found to be

And ultimately the equation of the meridonal curve of the parabolic dome is



4-2

(4.3)

Due to the way the parameters of the parabola were set out, it was expected that k is negative

as this is what gives the concave shape which is required for a roofing structure to serve the

purposes for which it was intended for.

In order to have N varying with Φ, x needs to be eliminated from the equation. The critical

link between x and Φ can be found in the fact that the meridonal stress acts tangentially to the

curve at any point. And the first derivative of the meridonal curve evaluated at any point x

which gives the tangent to the curve at the point in question. The derivative can then be

equated to forming the following relationship.

(4.4)

The derivative of equation (3.4) can be found using simple differentiation techniques and lead

to the derivative being

(4.5)

Using the relationship established in equation (3.5) and equation (3.6), it is possible to obtain

an explicit equation of x in terms of Φ

(4.6)

With equation (3.7), the x-coordinate of any point along the curve can be described in terms

of the phi value. Limits for the phi values will need to be introduced as the angle cannot run

from 0 through 90 degrees because a phi value of 90 implies a vertical tangent line to the

curve, which of course does not exist. The section of the curve which will be rotated about

the y-axis has the domain . Substitution of these boundary values into equation

(3.7) yield the following limits for phi denoted as and respectively.

(4.7. a)

(4.7. b)

Substituting equation (3.7) back into (3.2) eliminates the x value and leaves us with the stress

in terms of phi.



4-3

(4.8)

The self-weight (W) is the only remaining variable which needs to be defined in terms of phi

(or x as we have the relationship between them).

The surface area of the paraboloid of revolution was found by using equation (2.5) in

conjunction with equation (3.5). The equation was then integrated making use of a place

holding variable so that the limits of integration may be in terms of x. A basic substitution

was used to evaluate the integral. The resulting surface area in terms of x was found to be:

*

+ (4.9)

Substitution of equation (3.7) into (3.11) to eliminate x and get the surface area to be in terms

of Φ so as to be consistent with equation (3.9) to which it was eventually substituted into.

*

+ (4.10)

Combining equations (3.10), (3.11) and (3.12) led to the meridonal stress at any point along

the curve in terms of phi.

4 (4.11) Meridonal stress equation for parabolic dome

*

+ (4.11)

In order to determine the exact equation for hoop stresses (), the relationship developed by

(Ugural, 1981) found in equation (2.1) will be re-written with as the subject.

Equations (2.5), (2.6) and were first redefined in terms of phi using the relation in equation

(3.7). These equations along with equation (2.8) was used to eliminate and and from

the above equation.



4-4

After simplification the equation for the hoop stress was found to be

5 (4.12) Hoop stress equation for parabolic dome

*

+ (4.12)



5-1

5 Ellipsoidal Dome of Revolution

5.1 Method of Sections

Following the method of sections as described in chapter 3, the equation of the meridonal

curve for the ellipsoid of revolution can be defined by the equation

(5.1)

Or explicitly defined as

Where

: is the distance from the axis of revolution to curve at the base of the ellipsoid, this

will further be referred to as the radius of the ellipsoid.

: is the distance along the axis of revolution from the base of the parabola to its apex

and will further be referred to as the height of the ellipsoid.

: Defined, as the angle from the positive axis revolution to the normal of the curve.

Figure 5-1 Parameters of defined ellipsoidal meridian



5-2

The meridonal stress ( ) in the ellipsoidal dome under self-weight can be calculated using

equation (3.2).

(3.2)

In order to see the relationship of the meridonal stress with respect phi, will need to be

eliminated using the relationship:

(5.2)

Implicit differentiation was used to determine :

(5.3)

Substituting equation (5.3) into the relationship indicated in (5.2) allows to be explicitly

solved for in terms of phi.

(5.4)

With equation (5.4), the x-coordinate of any point along the curve can be described in terms

of the phi value; however it can be seen from equation (5.3) that at the point , there

exists a vertical tangent i.e. phi equals 90 degrees.

The loading on the ellipsoid () due to the self-weight of the structure at the point above the

cut can be written in terms of the unit weight of concrete, the thickness of the dome as well as

the surface area as presented in equation (2.8)

(2.8)

Using equations (2.8) and (5.4) to eliminate and from equation (3.2)

6 (5.5) Incomplete equilibrium equation for ellipsoidal dome

()

(5.5)

The surface area of the ellipsoid posed the greatest problem as it could not be expressed in

such a way that a relationship between phi and the surface area and subsequently the stresses

could be identified.



5-3

An alternative approach was considered.

5.2 The general stress equations for membrane theory

Making use of equations (2.1) and (2.2) the stress equations for the hoop and meridonal

stresses can be determined

N ∫ . (2.1)

(2.2)

Where

: Meridonal radius of curvature

: Hoop radius of curvature

: Radial loading component

: Tangential loading component

From equation (3.1) and (3.2), the curvatures and can be calculated, with equation (5.4)

being used to eliminate .

(5.6)

(5.7)

The loading components which were identified in chapter 3 is used in conjunction with the

curvatures found above in order to compute the stress variation within the ellipsoidal dome.

(3.6)

(3.7)

Substitution of the above equations into the general stress formula leads to:

( )

∫

(5.8)



5-4

Due to the complexity of the integral in equation (3.8), evaluation was done with the aid of an

online integral solver, Wolfram Alpha™.

Figure 5-2 Output of online integral calculator (Source: Wolfram Alpha)

The evaluated integral was then substituted back into equation (3.8). After some

simplification the equation meridonal stress was found to be:

7 (5.9) Invalid meridonal stress equation for ellipsoidal dome

( )

(5.9)

From the above equation, certain problems were identified;

The term in the denominator

The

term in both the numerator and the denominator

The in the numerator.

The ellipsoid was defined in such a way that phi equals zero degrees corresponds to the apex

of the ellipsoidal dome with coordinates and phi equals ninety degrees corresponds to

the base of the dome with coordinates .

If we go back and see the way was defined in terms of phi by looking at the equation

There exists a problem in the limits of phi due to the fact that at the point where , phi is

undefined, which is expected due to the vertical tangent existing there. However the lower

limit of phi can be defined as 0 corresponding to .

In equation (5.9) the constant of integration, needs to be eliminated via the substitution of

known values into the equation. The only point where the meridonal stress is known is at theapex and its value is zero due to the fact that no load exists above the point. Substituting



5-5

And

This causes the equation to be undefined due to the term being zero in the

denominator.

The second problem is anticipated when parameterising the equation; all the terms can be

written in some ratio of height to radius except the term . Therefore it will not be possible to

observe the correlations in the stress and the ratio.

The final problem is encountered when phi equals ninety degrees; this once again leads to the

stress equation being undefined due to the term in the numerator.

Had the equation for the meridonal stress been successfully developed, the next step wouldhave been to determine the hoop stress by making use of equations (2.2), (3.7), (5.6) and

(5.7).

.



6-1

6 Parametric Results

6.1 Parabolic Dome of Revolution

A non-dimensional parameter for the concrete shell was defined to be lambda (λ) which was

defined as the ratio of height-to-radius of the parabolic dome.

(6.1)

Equation (6.1) was substituted in equations (4.11) and (4.12) making sure that the only

variables in the equations are lambda and phi. After simplification, the term remained in

the numerator, the resultant meridonal and hoop stresses was then divided by this term which

effectively made the left hand side of the equations dimensionless.

* + (6.2)

*

+ (6.3)

From the above equations we can see that for shells of the same shape (same height-to-radius

ratio λ) the stress resultant in the shell is directly proportional to radius of the dome, and

therefore also directly proportional to the height, due to the fact that lambda remains

constant. Therefore doubling the height or radius of the dome will effectively double the

hoop and meridonal stresses in the structure. The thickness of the structure also has a direct

relationship with the stress resultants this is because increasing or decreasing the thickness by

a factor will result in proportional increase or decrease to the stress resultants.

6.2 Ellipsoid of revolution

A non-dimensional parameter for the concrete shell was defined to be psi (ψ) which was

defined as the ratio of height-to-radius of the elliptical dome.

(6.4)

Psi is then substituted into equation (5.9) in order eliminate all other variables except that of

phi and psi.



6-2

(6.5)

As seen in the terms indicated in the red circles above, it is not possible to rewrite them in

terms of psi whereas all other terms in the expression can be.

Due to the reasons outlined here, it will not be possible to conduct a parametric study for the

ellipsoidal dome of revolution.



7-1

7 Discussion of Results

7.1 Parabolic dome of revolution

Non-Dimensional stress variations and plotted against the meridonal angle, Φ for

various values of λ ranging from 0.5 to 3.0 which covers most practical domes. This covers

most practical cases of the parabolic dome. From equations (4.7), it can be seen that the dome

lies within the interval which, in terms of lambda is .

Therefore the stress resultant equations are only valid within this interval. The intervals for

each corresponding value of lambda can be seen in table 1 below

Table 1 Range of phi values defining parabolic dome

λ Ranges of Φ (°)

0.5 0-45

1 0-63.4

1.5 0-71.6

2 0-76

2.5 0-78.7

3 0-80.5

7.1.1 Meridonal stress

It can be seen that the hoop stress is directly proportional to the radius of the dome as well as

the thickness of the shell, therefore doubling of either of these parameters will double the

hoop stress in the dome.

For all values of lambda, the meridonal stress remains negative (compressive) throughout the

shell. Theoretically the stress at the apex of the dome is zero; however this cannot be seen in

the graph due to the fact that the meridonal stress resultant function cannot be defined at the

point. Keeping lambda constant, it can be seen that the meridonal stress increases (in

compression) in an almost exponential trend, which is expected as the self-weight gets

cumulatively higher as you move toward the base of the dome reaching peak stress at the



7-2

base of the dome. These peaks in terms of non-dimensional stresses can be found in the

table below.

Table 2 Peak meridonal stress for corresponding values of lambda

λ Peak Stresses

0.5 -0.862

1 -0.948

1.5 -1.196

2 -1.484

2.5 -1.789

3 -2.103

From the parameterised equation (6.2) it is evident that lambda is inversely proportional to

the meridonal stress. I.e. a decrease in lambda induces an increase in the stress resultant. This

is illustrated in figure 7.1 below where for example at a constant phi value of 45°; the

meridonal stresses for lambda equalling 3; 2.5; 2; 1.5; 1 and 0.5 the non-dimensional

meridonal stress is found to be -0.144; -0.172; -0.215; -0.287; -0.431 and -0.862 respectively.

However, due to the fact that a higher value of lambda increases the phi interval, the higher

meridonal stresses will be found at the base of slender parabolic domes ( ) as seen

when comparing the peak non-dimensionless meridonal stresses. Consider the case of lambda

equal to 0.5; the peak non-dimensional stress occurs at 45° and equals -0.862. Looking at

lambda equal to 3; the peak non-dimensional stress occurs at 80.5° and equals -2.103.

A graph showing how the peak non-dimensional stress varies with lambda was developed

seen in Figure 7.2. There exists a value of lambda where the peak stress at the base of the

dome is at a minimum and any increase or decrease in lambda will cause the peak stress to

increase. This minimum occurs at a lambda value of 0.6 and has a corresponding non-

dimensional stress value of -0.847.



7-3

Figure 7-1 Meridonal stress distribution in parabolic dome

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0 10 20 30 40 50 60 70 80 90

N ᶲ / γ t a

Φ(degrees)

Dimensionless (Nᶲ/γta) Graph for determining

meridonal stresses at any point along a parabolic dome

λ=3 λ=2.5 λ=2 λ=1.5 λ=1 λ=0.5



7-4

Figure 7-2 Peak meridonal stress variation with lambda

-3.00

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0 0.5 1 1.5 2 2.5 3 3.5

N ᶲ / γ t a

λ

Dimensionless graph of (Nᶲ/γta) for determination of peak meridonal

stress for values of λ



7-5

7.1.2 Hoop stress

It can be seen that the hoop stress is directly proportional to the radius of the dome aswell as

the thickness of the shell, therefore doubling of either of these parameters will double the

hoop stress in the dome.

For all values of lambda, the hoop stress remains positive (tensile) throughout the shell. .

Theoretically the stress at the apex of the dome is zero; however this cannot be seen in the

graph due to the fact that the hoop stress resultant function cannot be defined at the point.

For a constant value of lambda, the hoop stress follows an almost linear trend along the

profile of the curve. For higher values of lambda, there is a shallow gradient and the change

in stress resultant from the apex to the base is minimal, whereas for smaller values of lambda,

there is a steeper gradient and there is a considerable difference in stresses at the apex and

base.

For example for a lambda value of 0.5, the non-dimensional apex and base stresses are 0.5

and 0.57 respectively. Compare those values to a lambda value of 3, with non-dimensional

apex and base stresses 0.09 and 0.11 respectively. A possible explanation for this is that as

the dome becomes more slender, more of the stress gets transferred through meridonal action

rather than hoop action. The peak hoop stresses at the base of the dome for each value oflambda can be found in the table below.

Table 3 Peak hoop stress for corresponding values of lambda

λ Peak Stresses

0.5 0.569

1 0.310

1.5 0.214

2 0.163

2.5 0.131

3 0.110

As with meridonal stress, the hoop stress within the dome is also inversely proportional to

lambda. As lambda decreases, the hoop stresses throughout the entire range of shell increases.

This trend can be seen in figure 7.3 below. At any point along the shell profile for lambda

equal to 0.5, the non-dimensional hoop stress is considerably larger than at any corresponding

point along the profile of lambda equal to 3. The peak stress for values of lambda ranging



7-6

from 0 to 3 was graphed; the peak stresses follow a hyperbolic trend where they are

maximum at low values of lambda and minimum at high values.

Due to the weakness of concrete in tension reinforcement might be required. If one assumes

that a minimum tensile stress of 2 MPa will cause failure with in the dome, it is possible to

identify under which geometric conditions this will occur and where reinforcement will be

required. From the trends seen in the non-dimensional charts, for lambda values of 1.5 and

greater, tensile reinforcement will only be required for very large structures due to the low

values of hoop stresses.

For example, consider a lambda value of 0.5 for a structure with height and base of 81m and

162m respectively. It is possible to calculate from which point tensile reinforcement will be

required. The point at which the tensile stress exceeds 2 MPa occurs at a phi value of 20° and

continues through to a phi value 45°. The corresponding x values can be calculated by using

equation (4.6).



7-7

Figure 7-3 Hoop stress distribution in parabolic dome

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 10 20 30 40 50 60 70 80 90

N ᶱ / γ t a

(Φ)Degrees

Dimensionless (Nᶱ/γta) Graph for determining

hoop stresses at any point along a parabolic dome

λ=3 λ=2.5 λ=2 λ=1.5 λ=1 λ=0.5



7-8

Figure 7-4 Peak hoop stress variation with lambda

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 0.5 1 1.5 2 2.5 3 3.5

N ᶱ / γ t a

λ

Dimensionless graph of (Nᶱ/γta) for determination of peak hoop stress

for values of λ



8-1

8 Numerical Example

Design a 100m high, 50m wide parabolic dome with an assumed thickness of 0.5m

Calculate λ equation (6.1)

Plot non-dimensional meridonal and hoop stress charts for λ equals 2 (Appendix A)

Multiply non-dimensional values by to obtain stress resultants

The peak meridonal stress will be calculated by

The peak hoop stress will be calculated by

No tensile reinforcement will be required.



9-1

9 Conclusion and Recommendations

The hoop and meridonal stress of the parabolic dome has been expressed in terms of a

geometric parameter λ which enabled the study of stress distributions with in the dome. Many

challenges were encountered with the ellipsoidal dome and ultimately it was not possible to

express the equations in terms of a single governing geometric property.

Focussing on the parabolic investigation, it was found that both the hoop and meridonal

stresses are directly proportional to both the thickness and base radius of the structure and

any change in these variables will produce a proportional change in the stress resultants.

The meridonal stress within the dome was found to increase as lambda decreased, however,

greater values of lambda implied a greater structure which in turn leads to a greater range of

phi values. This allows the peak meridonal stresses for higher values of lambda to surpass

that of lower values of lambda.

The hoop stress with in the dome followed the same inversely proportional relationship as

that of meridonal stress. However in the case of the hoop stress, a lower value of lambda

shows that the hoop stress throughout the structure is greater than the hoop stress at any point

along the profile of a dome with a higher value of lambda.

The design charts were developed for the hoop and meridonal stress distribution within a

parabolic dome for lambda values ranging from 0.5 to 3. These charts can be used to obtain

the stresses at any point along the profile of the parabolic.

This study can be further developed by increasing the scope of the investigation to include

the effects of wind loading on the structure. Doing so will induce shear and bending forces in

the structure.

Further development of the uncompleted ellipsoidal stress equations will need to be further

developed in order to complete a parametric study on this shaped dome.



10-1

10 References

Baker, E. H., Kovalevsky, L., & Rish, F. L. (1972). Structural Analysis of Shells.

Farshad, M. (1992). Design and Analysis of Shell Structures.

Stewart, J. (2006). Calculus Concepts & Contexts 3.

Ugural, A. (1981). Stresses in Plates & Shells.

Zingoni, A. (1997). Shell Structures In Civil & Mechanical Engineering.

KRIVOSHAPKO, S.N., (2007). Research on General and Axisymmetric Ellipsoidal Shells

Used as Domes, Pressure Vessels, and Tanks.

Applied Mechanics Reviews,

60(6), pp. 336.

NEMENYI, P. and TRUESDELL, C., (1943). A Stress Function for the Membrane Theory of

Shells of Revolution.

Proceedings of the National Academy of Sciences of the United States of

America, 29(5), pp. pp. 159-162.

WAN, F.Y.M. and WEINITSCHKE, H.J., (1988). On shells of revolution with the Love-

Kirchhoff hypotheses.

Journal of Engineering Mathematics,

22(4), pp. 285-334.

ZINGONI, A., (2002). Parametric stress distribution in shell-of-revolution sludge digesters

of parabolic ogival form.

Thin-Walled Structures,

40(7 – 8), pp. 691-702.



i

Appendixes



ii

Appendix A: Design tables for

parabolic dome

Meridonal stresses

Φ(degrees)

1 -0.500

2 -0.500

3 -0.501

4 -0.502

5 -0.503

6 -0.504

7 -0.506

8 -0.507

9 -0.509

10 -0.512

11 -0.514

12 -0.517

13 -0.52014 -0.523

15 -0.527

16 -0.531

17 -0.535

18 -0.539

19 -0.544

20 -0.549

21 -0.555

22 -0.561

23 -0.567

24 -0.574

25 -0.581

26 -0.588

27 -0.596

28 -0.605

29 -0.614

30 -0.623

31 -0.633

32 -0.644

33 -0.655

34 -0.667

35 -0.680

36 -0.694

37 -0.70838 -0.723

39 -0.739

40 -0.757

41 -0.775

42 -0.795

43 -0.816

44 -0.838

45 -0.862



iii

Φ(degrees)

1 -0.2502 -0.250

3 -0.251

4 -0.251

5 -0.251

6 -0.252

7 -0.253

8 -0.254

9 -0.255

10 -0.256

11 -0.257

12 -0.258

13 -0.260

14 -0.262

15 -0.263

16 -0.265

17 -0.267

18 -0.270

19 -0.272

20 -0.275

21 -0.277

22 -0.280

23 -0.283

24 -0.287

25 -0.290

26 -0.294

27 -0.298

28 -0.302

29 -0.30730 -0.312

31 -0.317

32 -0.322

33 -0.328

34 -0.334

35 -0.340

36 -0.347

37 -0.354

38 -0.362

39 -0.370

40 -0.378

41 -0.388

42 -0.397

43 -0.408

44 -0.41945 -0.431

46 -0.444

47 -0.457

48 -0.472

49 -0.488

50 -0.505

51 -0.523

52 -0.543

53 -0.564

54 -0.587

55 -0.613

56 -0.640

57 -0.670

58 -0.702

59 -0.738

60 -0.778

61 -0.821

62 -0.870

63 -0.923

63.43 -0.948



iv

Φ(degrees)

1 -0.1672 -0.167

3 -0.167

4 -0.167

5 -0.168

6 -0.168

7 -0.169

8 -0.169

9 -0.170

10 -0.171

11 -0.171

12 -0.172

13 -0.173

14 -0.174

15 -0.176

16 -0.177

17 -0.178

18 -0.180

19 -0.181

20 -0.183

21 -0.185

22 -0.187

23 -0.189

24 -0.191

25 -0.194

26 -0.196

27 -0.199

28 -0.202

29 -0.20530 -0.208

31 -0.211

32 -0.215

33 -0.218

34 -0.222

35 -0.227

36 -0.231

37 -0.236

38 -0.241

39 -0.246

40 -0.252

41 -0.258

42 -0.265

43 -0.272

44 -0.27945 -0.287

46 -0.296

47 -0.305

48 -0.315

49 -0.325

50 -0.337

51 -0.349

52 -0.362

53 -0.376

54 -0.392

55 -0.408

56 -0.427

57 -0.447

58 -0.468

59 -0.492

60 -0.519

61 -0.548

62 -0.580

63 -0.616

64 -0.655

65 -0.700

66 -0.751

67 -0.808

68 -0.873

69 -0.947

70 -1.033

71 -1.132

71.57 -1.196



v

Φ(degrees)

1 -0.1252 -0.125

3 -0.125

4 -0.125

5 -0.126

6 -0.126

7 -0.126

8 -0.127

9 -0.127

10 -0.128

11 -0.129

12 -0.129

13 -0.130

14 -0.131

15 -0.132

16 -0.133

17 -0.134

18 -0.135

19 -0.136

20 -0.137

21 -0.139

22 -0.140

23 -0.142

24 -0.143

25 -0.145

26 -0.147

27 -0.149

28 -0.151

29 -0.15330 -0.156

31 -0.158

32 -0.161

33 -0.164

34 -0.167

35 -0.170

36 -0.173

37 -0.177

38 -0.181

39 -0.185

40 -0.189

41 -0.194

42 -0.199

43 -0.204

44 -0.21045 -0.215

46 -0.222

47 -0.229

48 -0.236

49 -0.244

50 -0.252

51 -0.262

52 -0.271

53 -0.282

54 -0.294

55 -0.306

56 -0.320

57 -0.335

58 -0.351

59 -0.369

60 -0.389

61 -0.411

62 -0.435

63 -0.462

64 -0.492

65 -0.525

66 -0.563

67 -0.606

68 -0.654

69 -0.710

70 -0.774

71 -0.849

72 -0.93673 -1.039

74 -1.162

75 -1.310

75.96 -1.484



vi

.5

Φ(degrees)

1 -0.1002 -0.100

3 -0.100

4 -0.100

5 -0.101

6 -0.101

7 -0.101

8 -0.101

9 -0.102

10 -0.102

11 -0.103

12 -0.103

13 -0.104

14 -0.105

15 -0.105

16 -0.106

17 -0.107

18 -0.108

19 -0.109

20 -0.110

21 -0.111

22 -0.112

23 -0.113

24 -0.115

25 -0.116

26 -0.118

27 -0.119

28 -0.121

29 -0.12330 -0.125

31 -0.127

32 -0.129

33 -0.131

34 -0.133

35 -0.136

36 -0.139

37 -0.142

38 -0.145

39 -0.148

40 -0.151

41 -0.155

42 -0.159

43 -0.163

44 -0.16845 -0.172

46 -0.177

47 -0.183

48 -0.189

49 -0.195

50 -0.202

51 -0.209

52 -0.217

53 -0.226

54 -0.235

55 -0.245

56 -0.256

57 -0.268

58 -0.281

59 -0.295

60 -0.311

61 -0.329

62 -0.348

63 -0.369

64 -0.393

65 -0.420

66 -0.450

67 -0.485

68 -0.524

69 -0.568

70 -0.620

71 -0.679

72 -0.74973 -0.831

74 -0.930

75 -1.048

76 -1.193

77 -1.372

78 -1.597

78.69 -1.789



vii

Φ(degrees)

1 -0.0832 -0.083

3 -0.084

4 -0.084

5 -0.084

6 -0.084

7 -0.084

8 -0.085

9 -0.085

10 -0.085

11 -0.086

12 -0.086

13 -0.087

14 -0.087

15 -0.088

16 -0.088

17 -0.089

18 -0.090

19 -0.091

20 -0.092

21 -0.092

22 -0.093

23 -0.094

24 -0.096

25 -0.097

26 -0.098

27 -0.099

28 -0.101

29 -0.10230 -0.104

31 -0.106

32 -0.107

33 -0.109

34 -0.111

35 -0.113

36 -0.116

37 -0.118

38 -0.121

39 -0.123

40 -0.126

41 -0.129

42 -0.132

43 -0.136

44 -0.14045 -0.144

46 -0.148

47 -0.152

48 -0.157

49 -0.163

50 -0.168

51 -0.174

52 -0.181

53 -0.188

54 -0.196

55 -0.204

56 -0.213

57 -0.223

58 -0.234

59 -0.246

60 -0.259

61 -0.274

62 -0.290

63 -0.308

64 -0.328

65 -0.350

66 -0.375

67 -0.404

68 -0.436

69 -0.473

70 -0.516

71 -0.566

72 -0.62473 -0.693

74 -0.775

75 -0.873

76 -0.994

77 -1.143

78 -1.331

79 -1.573

80 -1.890

80.54 -2.103



viii

Hoop stresses

Φ(degrees)

1 0.500

2 0.500

3 0.500

4 0.501

5 0.501

6 0.501

7 0.502

8 0.502

9 0.50310 0.504

11 0.505

12 0.505

13 0.506

14 0.507

15 0.508

16 0.510

17 0.511

18 0.512

19 0.513

20 0.515

21 0.516

22 0.518

23 0.520

24 0.521

25 0.523

26 0.525

27 0.527

28 0.529

29 0.531

30 0.533

31 0.535

32 0.537

33 0.539

34 0.541

35 0.544

36 0.546

37 0.54838 0.551

39 0.553

40 0.556

41 0.558

42 0.561

43 0.56444 0.566

45 0.569



ix

Φ(degrees)

1 0.2502 0.250

3 0.250

4 0.250

5 0.250

6 0.251

7 0.251

8 0.251

9 0.252

10 0.252

11 0.252

12 0.253

13 0.253

14 0.254

15 0.254

16 0.255

17 0.255

18 0.256

19 0.257

20 0.257

21 0.258

22 0.259

23 0.260

24 0.261

25 0.262

26 0.262

27 0.263

28 0.264

29 0.26530 0.266

31 0.267

32 0.268

33 0.270

34 0.271

35 0.272

36 0.273

37 0.274

38 0.275

39 0.277

40 0.278

41 0.279

42 0.281

43 0.282

44 0.28345 0.285

46 0.286

47 0.287

48 0.289

49 0.290

50 0.291

51 0.293

52 0.294

53 0.296

54 0.297

55 0.298

56 0.300

57 0.301

58 0.303

59 0.304

60 0.306

61 0.307

62 0.308

63 0.310

63.43 0.310



x

Φ(degrees)

1 0.1672 0.167

3 0.167

4 0.167

5 0.167

6 0.167

7 0.167

8 0.167

9 0.168

10 0.168

11 0.168

12 0.168

13 0.169

14 0.169

15 0.169

16 0.170

17 0.170

18 0.171

19 0.171

20 0.172

21 0.172

22 0.173

23 0.173

24 0.174

25 0.174

26 0.175

27 0.176

28 0.176

29 0.17730 0.178

31 0.178

32 0.179

33 0.180

34 0.180

35 0.181

36 0.182

37 0.183

38 0.184

39 0.184

40 0.185

41 0.186

42 0.187

43 0.188

44 0.18945 0.190

46 0.191

47 0.191

48 0.192

49 0.193

50 0.194

51 0.195

52 0.196

53 0.197

54 0.198

55 0.199

56 0.200

57 0.201

58 0.202

59 0.203

60 0.204

61 0.205

62 0.206

63 0.206

64 0.207

65 0.208

66 0.209

67 0.210

68 0.211

69 0.212

70 0.213

71.57 0.214



xi

Φ(degrees)

1 0.1252 0.125

3 0.125

4 0.125

5 0.125

6 0.125

7 0.125

8 0.126

9 0.126

10 0.126

11 0.126

12 0.126

13 0.127

14 0.127

15 0.127

16 0.127

17 0.128

18 0.128

19 0.128

20 0.129

21 0.129

22 0.129

23 0.130

24 0.130

25 0.131

26 0.131

27 0.132

28 0.132

29 0.13330 0.133

31 0.134

32 0.134

33 0.135

34 0.135

35 0.136

36 0.137

37 0.137

38 0.138

39 0.138

40 0.139

41 0.140

42 0.140

43 0.141

44 0.14245 0.142

46 0.143

47 0.144

48 0.144

49 0.145

50 0.146

51 0.146

52 0.147

53 0.148

54 0.149

55 0.149

56 0.150

57 0.151

58 0.151

59 0.152

60 0.153

61 0.153

62 0.154

63 0.155

64 0.156

65 0.156

66 0.157

67 0.158

68 0.158

69 0.159

70 0.159

71 0.160

72 0.16173 0.161

74 0.162

75 0.162

75.96 0.163



xii

Φ(degrees)

1 0.1002 0.100

3 0.100

4 0.100

5 0.100

6 0.100

7 0.100

8 0.100

9 0.101

10 0.101

11 0.101

12 0.101

13 0.101

14 0.101

15 0.102

16 0.102

17 0.102

18 0.102

19 0.103

20 0.103

21 0.103

22 0.104

23 0.104

24 0.104

25 0.105

26 0.105

27 0.105

28 0.106

29 0.10630 0.107

31 0.107

32 0.107

33 0.108

34 0.108

35 0.109

36 0.109

37 0.110

38 0.110

39 0.111

40 0.111

41 0.112

42 0.112

43 0.113

44 0.11345 0.114

46 0.114

47 0.115

48 0.115

49 0.116

50 0.117

51 0.117

52 0.118

53 0.118

54 0.119

55 0.119

56 0.120

57 0.121

58 0.121

59 0.122

60 0.122

61 0.123

62 0.123

63 0.124

64 0.124

65 0.125

66 0.125

67 0.126

68 0.127

69 0.127

70 0.128

71 0.128

72 0.12873 0.129

74 0.129

75 0.130

76 0.130

77 0.131

78 0.131

78.69 0.131



xiii

Φ(degrees)

1 0.0832 0.083

3 0.083

4 0.083

5 0.083

6 0.084

7 0.084

8 0.084

9 0.084

10 0.084

11 0.084

12 0.084

13 0.084

14 0.085

15 0.085

16 0.085

17 0.085

18 0.085

19 0.086

20 0.086

21 0.086

22 0.086

23 0.087

24 0.087

25 0.087

26 0.087

27 0.088

28 0.088

29 0.08830 0.089

31 0.089

32 0.089

33 0.090

34 0.090

35 0.091

36 0.091

37 0.091

38 0.092

39 0.092

40 0.093

41 0.093

42 0.094

43 0.094

44 0.09445 0.095

46 0.095

47 0.096

48 0.096

49 0.097

50 0.097

51 0.098

52 0.098

53 0.099

54 0.099

55 0.099

56 0.100

57 0.100

58 0.101

59 0.101

60 0.102

61 0.102

62 0.103

63 0.103

64 0.104

65 0.104

66 0.105

67 0.105

68 0.105

69 0.106

70 0.106

71 0.107

72 0.10773 0.107

74 0.108

75 0.108

76 0.108

77 0.109

78 0.109

79 0.109

80 0.110

80.54 0.110



xiv

Appendix B: Logbook

Date Comments

20/09 Collected Journals

Met with Prof Zingoni to arrange a meeting for 21/09

21/09 Met with Prof Zingoni

Told me to redefine stress equations in terms of sphericalcoordinates

22-24/09 Long weekend

25/09 Obtained Template for thesis document

Obtained masters thesis for referencing purposes

Made contact with Deon Solomons in the maths department

26/09 Continued work on deriving equations

Setup meeting with Dr Neil Roberston for 27/09 at 10:00am

27/09 Met with Dr Robertson

Told me to try cylindrical co-ordinates as the shapes are symmetrical

28/09 Formatted Thesis Document

Placed headings and sub-headings

Inserted comments

29-30/09 Weekend

1/10/2012 Continued work on derivation of equations

Emailed Prof Zingoni

2/10/2012 Continued work on derivation of equations

No reply from Prof Zingoni

3/10/2012 Continued work on derivations

Re-emailed Prof Zingoni



xv

In the space of 3 days, Naweed Kahaar received 2 responses fromProf Zingoni

4/10/2012 Created excel log book

5/10/2012 Arrangend meeting with Zingoni

6-7/10/2012 Weekend

8/10/2012 Meeting with Zingoni

8-14/10/2012 Continued work on derivations

15-21/10/2012 Prepared draft for submission

22/10/2012 Submitted draft

29/10/2012 Received feed back from draft

30/10-7/11 Attempted derivations of ellipsoidal shell

7-10/11/2012 Final editing

11/11/2012 Print and bound

12/11/2012 Submission

prbld thesis derivations

Documents