This document presents an undergraduate thesis analyzing the stress distribution in parabolic and ellipsoidal concrete domes. The student, Nazeer Slarmie, conducted a literature review on shell structures and shell theory. Slarmie then developed closed form solutions to determine the membrane stress equations for a parabolic dome of revolution and attempted to do the same for an ellipsoidal dome of revolution through a method of sections.
Node.js and Containers Go Together Like Peanut Butter and JellyRoss Kukulinski
Slides from Ross Kukulinski's (@rosskukulinski) presentation at Node Interactive Europe in September 2016. In this presentation, Ross Kukulinski, a Nodejs Evangelist and container enthusiast, will discuss why Node.js has become a popular choice in microservices-oriented, containerized dynamic environments. He will discuss how to get started with Node.js, Docker and Kubernetes and what pitfalls often occur when starting and how to avoid them.
In 2012, the U.S. solar industry installed 3.3 GW of solar capacity, growing 76% over 2011. What’s happened with solar in your state? It was a record year for many states. California became the first state to install over 1 GW in one year, with cumulative installations making it 7th in the world (if it were its own country). Arizona came in as the second largest market, spurred by installations of huge utility-scale solar power plants, while New Jersey, ranked third, was led by a growing solar industry, which numbers more than tanning salons in that state. Learn more at seia.org/smi
Design Genro plans to launch a new product line in 2014 and has developed a preliminary marketing plan. The plan focuses on market research to identify target customers, developing an advertising campaign to promote the new products, and determining the best pricing strategy to maximize sales. If successful, the plan would help Design Genro introduce its innovative new designs to more customers.
Node.js and Containers Go Together Like Peanut Butter and JellyRoss Kukulinski
Slides from Ross Kukulinski's (@rosskukulinski) presentation at Node Interactive Europe in September 2016. In this presentation, Ross Kukulinski, a Nodejs Evangelist and container enthusiast, will discuss why Node.js has become a popular choice in microservices-oriented, containerized dynamic environments. He will discuss how to get started with Node.js, Docker and Kubernetes and what pitfalls often occur when starting and how to avoid them.
In 2012, the U.S. solar industry installed 3.3 GW of solar capacity, growing 76% over 2011. What’s happened with solar in your state? It was a record year for many states. California became the first state to install over 1 GW in one year, with cumulative installations making it 7th in the world (if it were its own country). Arizona came in as the second largest market, spurred by installations of huge utility-scale solar power plants, while New Jersey, ranked third, was led by a growing solar industry, which numbers more than tanning salons in that state. Learn more at seia.org/smi
Design Genro plans to launch a new product line in 2014 and has developed a preliminary marketing plan. The plan focuses on market research to identify target customers, developing an advertising campaign to promote the new products, and determining the best pricing strategy to maximize sales. If successful, the plan would help Design Genro introduce its innovative new designs to more customers.
Este documento trata sobre los primeros auxilios. Explica que los primeros auxilios son la ayuda inmediata que se brinda a personas que han sufrido un accidente o crisis de salud hasta que reciban atención médica calificada. Los objetivos de los primeros auxilios son salvar vidas, evitar que las lesiones empeoren y favorecer la recuperación. También describe los pasos básicos para realizar un primer auxilio como hacer un reconocimiento de la víctima, controlar las vías respiratorias, buscar la respiración
Importancia de-la-tic-en-la-educacion1600Yoherly01
El documento discute las razones para usar tecnologías de la información y la comunicación (TIC) en la educación, incluyendo la alfabetización digital de los estudiantes, la productividad y la innovación en las prácticas docentes. También analiza las perspectivas de las TIC en la educación desde las perspectivas de los equipos, bibliotecas, materiales educativos, cursos de capacitación, aprendizaje, estudiantes, profesores y el ámbito laboral, social y familiar.
In the world of perimeter security the practice to see first, know first and act first is not always the organizations primary mission. Security Managers share some of their primary concerns to optimize their standard operating procedures.
O documento discute teorias sobre ensino e aprendizagem na educação superior, incluindo: 1) Novas teorias baseadas na psicologia cognitiva que enfatizam a aprendizagem centrada no estudante adulto; 2) A inteligência como um espectro de competências e as capacidades intelectuais humanas; 3) Avaliação dos estudantes com trabalhos escritos e práticos.
Inconfidência nº 233/Edição histórica da intentona comunista de1935Lucio Borges
Aí está mais uma edição histórica do jornal Inconfidência, acerca da hedionda 'Intentona Comunista', de novembro de 1935. Este muito doloroso episódio de nossa História é relembrado, todos os anos, junto ao 'Monumento Votivo' às vítimas dos covardes comunistas, na Praia Vermelha, no Rio de Janeiro, no dia 27 de novembro; o dito episódio pode ser sintetizado por duas palavras apenas: traição e covardia. E, como vem iterativamente afirmando o bravo jornal, "Esquecer também é Trair!"
Recomendo, em especial aos formadores de opinião, a leitura atenta e custódia das matérias das pág 6 - 'Discurso do Deputado Federal Afonso de Carvalho' e pág 25, 'Carta da Progenitora do Capitão Benedicto Bragança aos comunistas': a primeira, pelos profundos ensinamentos históricos que contém, na severa condenação de duas ideologias que sempre nos assolaram - o positivismo e o comunismo; e a segunda, pelo pungente sentimento de revolta de uma mãe, sobre a morte traiçoeira de seu jovem filho, na Escola de Aviação Militar.
Acesse o site: www.rsnoticias.net
- Archeology: before and without Kubernetes
- Deployment: kube-up, DCOS, GKE
- Core Architecture: the apiserver, the kubelet and the scheduler
- Compute Model: the pod, the service and the controller
This document summarizes the YAPC::Asia Tokyo 2012 conference. It discusses changes from the previous year including a new venue and introducing LT-thon sessions. Over 800 people attended, more than expected. The speaker thanked sponsors, staff, and attendees for making the conference a success. Attendee numbers have grown significantly each year. The speaker hopes future YAPC::Asia conferences will be even more welcoming and bring greater awareness to the community.
See also (Sorry, mainly in Japanese)
http://go-talks.appspot.com/github.com/lestrrat/go-slides/tree/master/2014-golangstudy-HDE
http://go-talks.appspot.com/github.com/lestrrat/go-slides/2014-yapcasia-go-for-perl-mongers/main.slide#1
Sustainability Assessment of Complex BridgesJames Serpell
This document provides an advanced final year project report on developing a sustainability assessment model for complex bridges. It begins with a literature review of existing sustainability assessment models for bridges. It then develops a new Complex Bridge Sustainability Assessment (CBSA) model by adapting and improving on a previous New Road Bridge Sustainability Assessment model. Two case studies are presented applying the CBSA model: the Oresund Link bridge and the proposed Solent Link bridge. The results of these assessments are analyzed and the CBSA model is critically evaluated against the previous model.
This document is the dissertation of Susovan Pal submitted to Rutgers University in partial fulfillment of the requirements for a Doctor of Philosophy degree in Mathematics. The dissertation contains two main parts. Part I studies the regularity properties of Douady-Earle extensions of circle homeomorphisms. It proves that Douady-Earle extensions of Holder continuous circle homeomorphisms are Holder continuous with the same exponent, and Douady-Earle extensions of circle diffeomorphisms are diffeomorphisms of the closed unit disk. Part II analyzes the eigenvalues of the Laplace operator acting on Riemann surfaces, showing that given any small positive number and natural number k, there exists a Riemann surface whose k
Modelling the seismic response of an unreinforced masonry structureWilco van der Mersch
The document describes a finite element analysis of an unreinforced masonry structure conducted as part of a master's thesis. The structure analyzed is based on a two-story terraced house tested as part of the ESECMaSE project. A finite element model is developed using solid elements and a smeared crack model. Eigenvalue and pushover analyses are carried out to model the structure's modal parameters and seismic response. The results are compared to the experimental tests to validate the model. Key findings include the model reasonably approximating the structure's mode shapes and frequencies, and the cyclic pushover analysis best matching the load-displacement response observed in the pseudo-dynamic test.
This document acknowledges and thanks those who supported the author's thesis work. It thanks the author's advisor, Professor Richard Royce, for guidance throughout the thesis. It also thanks Donald L. Blount and Associates and Accurate Pattern, Inc. for funding and constructing two hull models tested in the thesis. Lastly, it thanks Stevens Institute of Technology for use of their testing facilities and several individuals who provided assistance during tank testing.
This document is a bachelor thesis that investigates the structural mechanic FE-analysis of the nonlinear behavior of multilayer cables using beam elements in ANSYS. It begins with an introduction that describes multilayer high voltage cables and the need to simulate their assembly processes. The thesis then reviews previous related works and provides theoretical background on structural mechanics, finite element analysis, friction modeling, and ANSYS settings. It presents simulations that validate friction mapping between beam and solid element models to verify the correct representation of isotropic and static-dynamic friction. Finally, the thesis simulates a non-twisted high voltage cable and discusses prospects for future work in fully simulating twisted cables.
The document summarizes Robert Murtagh's PhD thesis on analytical models of single bubbles and foams. It includes a declaration of authorship, acknowledgements, and a summary of the thesis contents. The thesis uses a "Z-cone model" to model bubbles as collections of circular cones joined at their bases. It applies this model to study the energy of ordered foams and the bubble-bubble interaction over a range of liquid fractions. It also adapts the model to study the energy of a Kelvin foam cell and investigates contact losses in the Kelvin structure away from the wet limit. Finally, it examines the evolution of gas bubbles on a liquid surface containing mixtures of gases with different solubilities.
Behaviour and Analysis of Large Diameter Laterally Loaded PilesHenry Pik Yap Sia
75% of UK offshore wind turbines are supported on monopile foundations (Doherty and Gavin, 2012). The piles are subjected to large lateral loading from wind and tide surges as well as seabed movement. British Standards (BS EN 61400-3:2009) suggested p-y curve to predict the behaviour of laterally loaded offshore piles. P-y curve has certain assumptions including negligible rotational resistance along the pile length.
This report presents our investigation on the effect of rotational resistance on a typical large diameter pile. It also describes how the finite difference (FD) program has been written from first principles, the Winkler’s Method and Euler-Bernoulli Beam theory. To calculate the rotational resistance, the slice method proposed by McVay and Niraula (2004) is implemented in our model. Our linear-elastic FD model calculates the displacement, bending moment, shear force and soil pressure for laterally loaded piles for two cases: (a) when rotational resistance is considered and (b) when rotational resistance is neglected. The later represents the values used in the industry.
Sensitivity study, through our model produced good results within its scope. The results suggested that the change in the soil and pile properties was found to be dependent on the length-to-depth (L/D) ratio of the pile and the stiffness of the soil next to the pile. In other words, when reached critical ratio, the rotational resistance becomes very significant, specifically for short, rigid piles. Therefore, we computed curves to recommend the range of L/D values where rotational resistance can be safely neglected.
Recommendations and suggestions are made to improve the model and research to fully encapsulate the behaviour of offshore monopiles, such as cyclic loading, elastic continuum, plasticity and non-linearity.
Lastly, we have sufficient confidence from this research to conclude that rotational resistance of a laterally loaded large diameter pile are important and that current design standards for offshore monopiles are conservative.
This document analyzes particle fracture in hypereutectic aluminum-silicon alloys through Vickers microindentation tests. Results for second-phase silicon particles were compared to monolithic silicon. Subsurface damage mechanisms beneath indented silicon particles were studied to determine the crack morphologies causing particle fracture and the role of indentation-induced phase transformations. Cross-sectional FIB and TEM revealed a semi-circular plastic core and subsurface lateral crack pattern below residual indents, in addition to a localized amorphous silicon zone below the plastic core at crack edges.
Este documento trata sobre los primeros auxilios. Explica que los primeros auxilios son la ayuda inmediata que se brinda a personas que han sufrido un accidente o crisis de salud hasta que reciban atención médica calificada. Los objetivos de los primeros auxilios son salvar vidas, evitar que las lesiones empeoren y favorecer la recuperación. También describe los pasos básicos para realizar un primer auxilio como hacer un reconocimiento de la víctima, controlar las vías respiratorias, buscar la respiración
Importancia de-la-tic-en-la-educacion1600Yoherly01
El documento discute las razones para usar tecnologías de la información y la comunicación (TIC) en la educación, incluyendo la alfabetización digital de los estudiantes, la productividad y la innovación en las prácticas docentes. También analiza las perspectivas de las TIC en la educación desde las perspectivas de los equipos, bibliotecas, materiales educativos, cursos de capacitación, aprendizaje, estudiantes, profesores y el ámbito laboral, social y familiar.
In the world of perimeter security the practice to see first, know first and act first is not always the organizations primary mission. Security Managers share some of their primary concerns to optimize their standard operating procedures.
O documento discute teorias sobre ensino e aprendizagem na educação superior, incluindo: 1) Novas teorias baseadas na psicologia cognitiva que enfatizam a aprendizagem centrada no estudante adulto; 2) A inteligência como um espectro de competências e as capacidades intelectuais humanas; 3) Avaliação dos estudantes com trabalhos escritos e práticos.
Inconfidência nº 233/Edição histórica da intentona comunista de1935Lucio Borges
Aí está mais uma edição histórica do jornal Inconfidência, acerca da hedionda 'Intentona Comunista', de novembro de 1935. Este muito doloroso episódio de nossa História é relembrado, todos os anos, junto ao 'Monumento Votivo' às vítimas dos covardes comunistas, na Praia Vermelha, no Rio de Janeiro, no dia 27 de novembro; o dito episódio pode ser sintetizado por duas palavras apenas: traição e covardia. E, como vem iterativamente afirmando o bravo jornal, "Esquecer também é Trair!"
Recomendo, em especial aos formadores de opinião, a leitura atenta e custódia das matérias das pág 6 - 'Discurso do Deputado Federal Afonso de Carvalho' e pág 25, 'Carta da Progenitora do Capitão Benedicto Bragança aos comunistas': a primeira, pelos profundos ensinamentos históricos que contém, na severa condenação de duas ideologias que sempre nos assolaram - o positivismo e o comunismo; e a segunda, pelo pungente sentimento de revolta de uma mãe, sobre a morte traiçoeira de seu jovem filho, na Escola de Aviação Militar.
Acesse o site: www.rsnoticias.net
- Archeology: before and without Kubernetes
- Deployment: kube-up, DCOS, GKE
- Core Architecture: the apiserver, the kubelet and the scheduler
- Compute Model: the pod, the service and the controller
This document summarizes the YAPC::Asia Tokyo 2012 conference. It discusses changes from the previous year including a new venue and introducing LT-thon sessions. Over 800 people attended, more than expected. The speaker thanked sponsors, staff, and attendees for making the conference a success. Attendee numbers have grown significantly each year. The speaker hopes future YAPC::Asia conferences will be even more welcoming and bring greater awareness to the community.
See also (Sorry, mainly in Japanese)
http://go-talks.appspot.com/github.com/lestrrat/go-slides/tree/master/2014-golangstudy-HDE
http://go-talks.appspot.com/github.com/lestrrat/go-slides/2014-yapcasia-go-for-perl-mongers/main.slide#1
Sustainability Assessment of Complex BridgesJames Serpell
This document provides an advanced final year project report on developing a sustainability assessment model for complex bridges. It begins with a literature review of existing sustainability assessment models for bridges. It then develops a new Complex Bridge Sustainability Assessment (CBSA) model by adapting and improving on a previous New Road Bridge Sustainability Assessment model. Two case studies are presented applying the CBSA model: the Oresund Link bridge and the proposed Solent Link bridge. The results of these assessments are analyzed and the CBSA model is critically evaluated against the previous model.
This document is the dissertation of Susovan Pal submitted to Rutgers University in partial fulfillment of the requirements for a Doctor of Philosophy degree in Mathematics. The dissertation contains two main parts. Part I studies the regularity properties of Douady-Earle extensions of circle homeomorphisms. It proves that Douady-Earle extensions of Holder continuous circle homeomorphisms are Holder continuous with the same exponent, and Douady-Earle extensions of circle diffeomorphisms are diffeomorphisms of the closed unit disk. Part II analyzes the eigenvalues of the Laplace operator acting on Riemann surfaces, showing that given any small positive number and natural number k, there exists a Riemann surface whose k
Modelling the seismic response of an unreinforced masonry structureWilco van der Mersch
The document describes a finite element analysis of an unreinforced masonry structure conducted as part of a master's thesis. The structure analyzed is based on a two-story terraced house tested as part of the ESECMaSE project. A finite element model is developed using solid elements and a smeared crack model. Eigenvalue and pushover analyses are carried out to model the structure's modal parameters and seismic response. The results are compared to the experimental tests to validate the model. Key findings include the model reasonably approximating the structure's mode shapes and frequencies, and the cyclic pushover analysis best matching the load-displacement response observed in the pseudo-dynamic test.
This document acknowledges and thanks those who supported the author's thesis work. It thanks the author's advisor, Professor Richard Royce, for guidance throughout the thesis. It also thanks Donald L. Blount and Associates and Accurate Pattern, Inc. for funding and constructing two hull models tested in the thesis. Lastly, it thanks Stevens Institute of Technology for use of their testing facilities and several individuals who provided assistance during tank testing.
This document is a bachelor thesis that investigates the structural mechanic FE-analysis of the nonlinear behavior of multilayer cables using beam elements in ANSYS. It begins with an introduction that describes multilayer high voltage cables and the need to simulate their assembly processes. The thesis then reviews previous related works and provides theoretical background on structural mechanics, finite element analysis, friction modeling, and ANSYS settings. It presents simulations that validate friction mapping between beam and solid element models to verify the correct representation of isotropic and static-dynamic friction. Finally, the thesis simulates a non-twisted high voltage cable and discusses prospects for future work in fully simulating twisted cables.
The document summarizes Robert Murtagh's PhD thesis on analytical models of single bubbles and foams. It includes a declaration of authorship, acknowledgements, and a summary of the thesis contents. The thesis uses a "Z-cone model" to model bubbles as collections of circular cones joined at their bases. It applies this model to study the energy of ordered foams and the bubble-bubble interaction over a range of liquid fractions. It also adapts the model to study the energy of a Kelvin foam cell and investigates contact losses in the Kelvin structure away from the wet limit. Finally, it examines the evolution of gas bubbles on a liquid surface containing mixtures of gases with different solubilities.
Behaviour and Analysis of Large Diameter Laterally Loaded PilesHenry Pik Yap Sia
75% of UK offshore wind turbines are supported on monopile foundations (Doherty and Gavin, 2012). The piles are subjected to large lateral loading from wind and tide surges as well as seabed movement. British Standards (BS EN 61400-3:2009) suggested p-y curve to predict the behaviour of laterally loaded offshore piles. P-y curve has certain assumptions including negligible rotational resistance along the pile length.
This report presents our investigation on the effect of rotational resistance on a typical large diameter pile. It also describes how the finite difference (FD) program has been written from first principles, the Winkler’s Method and Euler-Bernoulli Beam theory. To calculate the rotational resistance, the slice method proposed by McVay and Niraula (2004) is implemented in our model. Our linear-elastic FD model calculates the displacement, bending moment, shear force and soil pressure for laterally loaded piles for two cases: (a) when rotational resistance is considered and (b) when rotational resistance is neglected. The later represents the values used in the industry.
Sensitivity study, through our model produced good results within its scope. The results suggested that the change in the soil and pile properties was found to be dependent on the length-to-depth (L/D) ratio of the pile and the stiffness of the soil next to the pile. In other words, when reached critical ratio, the rotational resistance becomes very significant, specifically for short, rigid piles. Therefore, we computed curves to recommend the range of L/D values where rotational resistance can be safely neglected.
Recommendations and suggestions are made to improve the model and research to fully encapsulate the behaviour of offshore monopiles, such as cyclic loading, elastic continuum, plasticity and non-linearity.
Lastly, we have sufficient confidence from this research to conclude that rotational resistance of a laterally loaded large diameter pile are important and that current design standards for offshore monopiles are conservative.
This document analyzes particle fracture in hypereutectic aluminum-silicon alloys through Vickers microindentation tests. Results for second-phase silicon particles were compared to monolithic silicon. Subsurface damage mechanisms beneath indented silicon particles were studied to determine the crack morphologies causing particle fracture and the role of indentation-induced phase transformations. Cross-sectional FIB and TEM revealed a semi-circular plastic core and subsurface lateral crack pattern below residual indents, in addition to a localized amorphous silicon zone below the plastic core at crack edges.
Flaps with Wavy Leading Edges for Robust Performance agains Upstream Trailing...Rafael Perez Torro
This document summarizes a master's dissertation that investigates the use of wavy leading edges on flaps to improve performance against upstream trailing vortices. Computational fluid dynamics simulations were performed using URANS turbulence models to test a two-element airfoil with wavy versus straight leading edge flaps. Initial studies optimized the relative position of the airfoil elements. Additional simulations then evaluated wavy flaps with varying amplitudes and wavelengths under clean and vortex conditions. Results showed wavy flaps performed worse than straight flaps without vortices but were more robust in the presence of upstream vortices.
This dissertation examines modeling and simulation of high pressure composite cylinders for hydrogen storage. It presents three papers on this topic:
1. The first paper develops a finite element model and neural network model to predict failure pressures of composite cylinders under mechanical and thermal loads.
2. The second paper extends the finite element model to analyze cylinders subjected to localized flame impingements, considering temperature-dependent properties and different failure modes.
3. The third paper studies fracture behavior of cylinders with an axial surface crack in the liner using fracture mechanics and accounts for the autofrettage effect.
This dissertation presents research on modeling and simulation of high pressure composite cylinders for hydrogen storage. It is divided into three parts. The first part develops a finite element model and neural network model to predict failure pressures of composite cylinders under mechanical and thermal loads. The second part models the behavior of cylinders subjected to flame impingement, including thermal damage and mechanical degradation. The third part studies fracture behavior of cylinders with an axial liner crack under various autofrettage levels using fracture mechanics and a global-local finite element approach.
1. The document discusses the design of a twisting skyscraper envelope through geometric modeling and structural optimization.
2. Different panelization approaches are explored through geometric algorithms to achieve developable surface geometries that allow for flat panels between floors.
3. A parametric modeling approach is used to generate potential base shapes and panel layouts. Structural performance is optimized through an iterative process to minimize weight.
01 - Publication - A use friendly Mutliscale lung modeling suite and applicat...Viraj Shah
This document describes the development of a user-friendly computational lung modeling suite to simulate gas transport. The lung model extends previous versions by incorporating a graphical user interface based on QT. The model accounts for the hierarchical branching structure of lungs down to the level of respiratory units called acini. The suite allows modifying structural properties related to lung diseases and aging, and executing functional simulations. As an application, the model is used to predict changes in oxygen uptake and gas concentration profiles during breathing with increasing severity of emphysema, a lung disease characterized by abnormal enlargement of airspaces. The results demonstrate the ability of the model to predict loss of lung function with disease progression.
This document is a dissertation submitted by Kwanda Tartibu for the degree of Master of Technology in Mechanical Engineering. It presents a simplified analysis of the vibration of variable length blades that could be used in wind turbine systems. Finite element models are developed in MATLAB and Unigraphics NX5 to calculate the natural frequencies of uniform, stepped and variable length beams. Experimental modal analysis is also performed and the results are compared to the numerical models. The effects of blade length and rotation on the natural frequencies are investigated.
Modeling and simulation of droplet dynamics for microfluidic applications 683...nataliej4
The document discusses a Creative Commons license that allows free use of the work under certain conditions, including requiring proper attribution of the original author and not allowing commercial use or modifications of the work. The license is a summary of the legal code for non-commercial sharing and distribution with attribution required and no derivatives allowed.
FZ4003 - Masters Dissertation - Synthesis of 4-(4-Alkylphenyl) and 4-(4-Alkyl...Theodore Hester
This document presents the results of research aimed at synthesizing cyclopent-3-ene-1,2-
dione and 4-(4-ethylphenyl)- and 4-(4-methoxyphenyl)-cyclopent-3-ene-1,2-diones as
potential ligands for metal complexes. While qualitative evidence supported the successful
synthesis of cyclopent-3-ene-1,2-dione, difficulties in separation meant only small quantities
could be obtained for further study. Synthesis of the extended ligands proved challenging
due to differences in reactivity of starting materials ethylbenzene and anisole, as well as
selectivity issues with the reactions attempted. Overall, the
This document is a project report submitted by students to fulfill the requirements of a course on mechanical engineering. It discusses the design and construction of a universal coupling. The report includes an acknowledgement, abstract, table of contents, and chapters on the literature review, design process, manufacturing process, and conclusions. The goal was to solve a problem related to designing a universal coupling to allow shaft misalignment and transmit torque, and determine the safe torque load. Relevant concepts, calculations, and the final manufactured universal coupling are described.
Numerical Investigation of Scouring at the Base of a Circular Pile in a Stead...Mark Donnelly-Orr
This thesis investigates scouring around the base of a circular pile exposed to steady tidal currents through numerical simulation. The study aims to determine the extent of scouring around wind turbine piles installed in Dublin Bay for the Dublin Array project and how scour is affected by current speed. Additional models assess scour prevention devices designed to disrupt vortices causing scour. Results show that scour initially occurs locally at currents of 0.225-0.275 m/s and increases in extent with current speed. Live-bed scour where the entire seabed is in motion occurs at 0.4-0.6 m/s. Scour prevention devices effectively disrupt vortices and reduce scouring. The main
The document describes a thesis that investigates the heat affected zone (HAZ) microstructure of BlastAlloy 160 (BA-160), a steel developed for blast-resistant naval applications. Various techniques including optical microscopy, SEM, APT, and microhardness testing were used to characterize the evolution of Cu precipitates, carbides, and martensite morphology across different HAZ regions simulated using a Gleeble thermal simulator. The results provide insights into the weldability of BA-160 by understanding the microstructural changes that occur in the HAZ during welding.
1. 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. Zingoni
Prepared by: Nazeer Slarmie
Student number: SLRMOG001
Date: 12th
November 2012
Thesis in partial fulfilment of the requirements for the degree of
BSc (Eng) Civil Engineering
2. 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:
3. 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.
4. 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
5. 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.
6. 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
7. 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
8. 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
9. 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
11. 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
12. 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
13. 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
14. 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
15. 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 beneficial
mechanical 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
16. 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
17. 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
18. 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
19. 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 of
the 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 middle
surface.
20. 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
21. 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,
22. 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
23. 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 Shells
The 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 is
zero 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.
24. 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)
25. 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 the
shell. 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)
26. 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)
27. 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.
28. 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 directions
respectively. 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
29. 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.
30. 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 N can 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.
31. 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
32. 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.
33. 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.
34. 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)
35. 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
36. 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.
37. 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)
38. 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 the
apex and its value is zero due to the fact that no load exists above the point. Substituting
39. 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 would
have been to determine the hoop stress by making use of equations (2.2), (3.7), (5.6) and
(5.7).
.
40. 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.
41. 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.
42. 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
43. 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.
44. 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
Nᶲ/γta
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
45. 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
Nᶲ/γta
λ
Dimensionless graph of (Nᶲ/γta) for determination of peak meridonal
stress for values of λ
46. 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 of
lambda 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
47. 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).
48. 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
Nᶱ/γta
(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
49. 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
Nᶱ/γta
λ
Dimensionless graph of (Nᶱ/γta) for determination of peak hoop stress
for values of λ
50. 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.
51. 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.
52. 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.
66. 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 spherical
coordinates
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
67. xv
In the space of 3 days, Naweed Kahaar received 2 responses from
Prof 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