DESIGN & CONSTRUCTION CRITERIA FOR DOMES IN LOW-COST HOUSING G. Talocchino A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, in fulfillment of the requirements for the degree of Master of Science in Engineering. Johannesburg, 2005 TABLE OF CONTENTS TABLE OF CONTENTS II DECLARATION I ABSTRACT II ACKNOWLEDGEMENTS III LIST OF FIGURES IV LIST OF TABLES IX LIST OF SYMBOLS XI 1. INTRODUCTION ....................................................................................................... 1 1.1 STATEMENT OF THE PROBLEM................................................................................. 1 1.2 AIMS OF THE INVESTIGATION................................................................................... 2 1.3 METHOD OF INVESTIGATION .................................................................................... 2 1.4 A BRIEF HISTORY OF THE DOME ............................................................................. 5 1.5 RECENT DEVELOPMENTS IN COMPRESSED EARTH & DOME............................... 7 CONSTRUCTION................................................................................................................. 7 1.5.1 FIBRE REINFORCED SOIL CRETE BLOCKS FOR THE CONSTRUCTION OF LOW- COST HOUSING ? RODRIGO FERNANDEZ (UNIVERSITY OF THE WITWATERSRAND, 2003) 7 1.5.2 DOMES IN LOW-COST HOUSING ? S.J. MAGAIA (UNIVERSITY OF THE WITWATERSRAND, SEPTEMBER 2003) 8 1.5.3 THE MONOLITHIC DOME SOLUTION 9 1.5.4 THE DOME SPACE SOLUTION 9 1.5.5 THE TLHOLEGO ECO-VILLAGE 10 1.6 ARCHITECTURAL CONSIDERATIONS...................................................................... 10 1.6.1 SHAPE 11 1.6.2 STRUCTURAL STRENGTH 13 1.6.3 RAIN PENETRATION & DAMP PROOFING 13 1.6.4 THERMAL PERFORMANCE 15 1.6.5 LIGHTING 16 1.6.6 INTERNAL ARRANGEMENT 16 2. SHAPE INVESTIGATION...................................................................................... 18 2.1 METHODS OF SHAPE INVESTIGATION ................................................................... 18 2.2 CLASSICAL THIN SHELL THEORY.......................................................................... 18 2.2.1 THE STRUCTURAL ELEMENTS 18 2.2.2 FORCES AND MOMENTS IN THE STRUCTURE 19 2.2.3 CLASSICAL DOME THEORY 20 2.2.4 CLASSICAL DOME/RING THEORY 22 DOME/RING THEORY IS THE THEORY APPLIED TO DOME STRUCTURES WITH A RING BEAM AT THE BASE OF THE STRUCTURE. THE STRUCTURE DOES NOT BEHAVE AS IF IT WERE FIXED OR PINNED, BUT SOMEWHERE BETWEEN THESE TWO CONDITIONS. 22 THE RING BEAM WILL ALLOW A CERTAIN AMOUNT OF ROTATION AND OUTWARD DISPLACEMENT AT THE BASE, WHICH A FIXED BASE WOULD NOT. THEREFORE, TWO NEW ERRORS ARE INTRODUCED AND THESE ARE ADDED TO THE DOME MEMBRANE ERRORS TO OBTAIN THE TOTAL DOME/RING ERRORS. 22 Ebd Nr d ey D R ?? ')12(cos 2 2 0 10 += (2.11) 22 2.3 BASIC FINITE ELEMENT ANALYSIS (FEA) THEORY............................................ 23 2.3.1 TYPES OF ELEMENTS USED 24 2.3.2 METHODS OF IMPROVING FEA ACCURACY 25 2.4 SENSITIVITY ANALYSIS............................................................................................ 25 2.4.1 DESCRIPTION OF THE ANALYSIS 25 2.4.2 SENSITIVITY ANALYSIS RESULTS 27 2.4.3 EFFECT OF VARYING THE STIFFNESS OF THE STRUCTURE 29 2.4.4 SUMMARY OF SENSITIVITY ANALYSIS FINDINGS 32 2.5 SHAPE ANALYSIS................................................................................................. 33 2.5.1 DESCRIPTION OF THE ANALYSIS 33 2.5.2 SHAPE EVALUATION CRITERIA 35 2.5.3 PRESENTATION OF THE ANALYSIS RESULTS 35 2.6 SHAPE ANALYSIS RESULTS (STRUCTURE TYPES A & C) .................................... 37 2.6.1 THE CATENARY 37 2.6.2 SECTIONS THROUGH THE HEMISPHERE 42 2.6.3 SECTIONS THROUGH THE PARABOLA 45 2.6.4 THE ELLIPSE 49 2.7 SHAPE ANALYSIS RESULTS (STRUCTURE TYPE B) .............................................. 52 2.8 SUMMARY OF THE RESULTS.................................................................................... 54 2.8.1 TABULATED SUMMARY 54 2.8.2 SUMMARY DISCUSSION 56 3. MATERIALS INVESTIGATION .......................................................................... 57 3.1 LITERATURE INVESTIGATION................................................................................. 57 3.1.1 CEMENT STABILIZED EARTH BLOCKS (CEB?S) 57 3.2 LABORATORY TESTING ........................................................................................... 59 3.2.1 FIBRE REINFORCED PLASTER TESTS 59 3.2.2 BRICKFORCE, WIRE WRAPPING & WIRE MESH TESTS 65 3.2.3 PLASTIC DAMP PROOF COURSE (DPC) FRICTION TESTS 68 4. STRUCTURAL ANALYSIS.................................................................................... 75 4.1 THE MODEL............................................................................................................... 76 4.1.1 THE DIMENSIONS OF THE STRUCTURE 76 4.1.2 THE FINITE ELEMENT ANALYSIS (FEA) MODEL 76 4.2 LOADING CALCULATIONS ....................................................................................... 77 4.2.1 DEAD LOAD (SELF WEIGHT) 77 4.2.2 LIVE LOAD 78 4.2.3 WIND LOAD 78 4.2.4 TEMPERATURE LOAD 80 4.3 LOAD COMBINATIONS.............................................................................................. 83 4.4 FINITE ELEMENT ANALYSIS RESULTS................................................................... 83 4.4.1 THE EFFECTS OF THE SKYLIGHT OPENINGS AND POINT LOADING 84 4.4.2 THE EFFECTS OF WINDOW AND DOOR OPENINGS 86 4.4.3 THE EFFECTS OF TEMPERATURE LOADING 90 4.4.4 FINAL DESIGN RESULTS 93 .5. DESIGN OF STRUCTURAL ELEMENTS....................................................... 103 5.1 DESIGN THEORY ..................................................................................................... 103 5.1.1 MASONRY DESIGN 103 5.1.2 FOUNDATION DESIGN 112 5.2 DESIGN CALCULATION RESULTS ......................................................................... 113 5.2.1 THE DOME 113 5.2.2 THE CYLINDER WALL 120 5.2.3 THE FOUNDATION DESIGN RESULTS 124 5.2.5 OVERALL STABILITY 125 6. CONSTRUCTION AND COST ANALYSIS ...................................................... 126 6.1 REINFORCED CONCRETE DOME CONSTRUCTION ............................................. 126 6.2 BRICK DOME AND VAULT CONSTRUCTION ........................................................ 127 6.3 CONSTRUCTION PROCEDURE OF THE PROTOTYPE 28M2 DOME...................... 129 6.3.1 SITE PREPARATION & SETTING OUT 129 6.3.2 THE FOUNDATION 130 6.3.3 THE CYLINDER WALL 132 6.3.4 THE INFLATABLE FORMWORK 133 6.3.5 THE DOME CONSTRUCTION 134 6.3.6 THE ARCHES AND SKYLIGHT CONSTRUCTION 135 6.3.7 WIRE WRAPPING AROUND THE OPENINGS 137 6.3.8 PLASTERING, PAINTING & FINISHING OF THE DOME 138 6.4 CONSTRUCTION MATERIALS INVESTIGATION ................................................... 139 6.4.1 MORTAR STRENGTH TESTS 139 6.4.2 FOUNDATION AND FLOOR SLAB TESTS 141 6.5 COST ANALYSIS ...................................................................................................... 143 7. CONCLUSIONS & RECOMMENDATIONS.................................................... 146 APPENDIX A (SELECTED ABAQUSTM OUTPUT) ............................................ 151 APPENDIX B (ALTERNATIVE STRUCTURE) .................................................. 154 REFERENCES............................................................................................................. 158 i DECLARATION I declare that this dissertation is my own, unaided work. It is being submitted for the Degree of Master of Science in Engineering in the University of the Witwatersrand, Johannesburg. It has not been submitted before for any degree or examination in any other University. Signature ____________________________ Date: 8th day of December 2005 ii ABSTRACT This dissertation investigates the different design and construction considerations involved when building a masonry dome. A detailed shape investigation was undertaken in order to summarize the best shaped dome structures. General recommendations are given for the shapes that produce the least tension and the most useable space. The effects of openings, temperature loading and wind loading were considered and a finite element analysis of the final structure was undertaken. It was found that regions of high tension exist around openings, especially under temperature loading, and materials suitable to resisting this tension were investigated (fibre plaster, chicken wire mesh and wire wrapping around openings). The final structure was built using an inflatable formwork. The construction procedure was well documented and a study of alternative methods of construction is presented. This dissertation shows that a durable, architecturally and structurally efficient low-cost masonry dome can be built if proper attention is given to minimizing and resisting tension within the structure. iii Acknowledgements I would like to thank Prof. M. Gohnert for his invaluable support and encouragement which made this dissertation possible. I am also grateful to HydraForm South Africa who supplied the cement stabilized earth blocks used in the construction of the dome, as well as the Central Johannesburg College who provided a site for the dome. The final dome structure was constructed by Dome Space South Africa. I would like to thank them for their involvement. Gianfranco Talocchino iv List of Figures CHAPTER 1 Figure 1.1 ? Distribution of South African Houses (RSA Census, 2001) pg1 Figure 1.2 ? Flow Diagram of the Investigation pg4 Figure 1.3 - Zulu Hut (KwaZulu-Natal) pg5 Figure 1.4 - Musgum Farmstead pg5 Figure 1.5 - The Ramesseum Storage Vaults, Gourna, Egypt pg6 Figure 1.6 - Low-Cost Dome built by S.J. Magaia(2003) pg8 Figure 1.7 ? EcoShells in Haiti. (www.monolithic.com) pg9 Figure 1.8 ? Dome Homes built by Dome Space pg9 Figure 1.9 ? Dome Cookhouse ? Tlholego Eco-Village pg10 Figure 1.10 ? Useable Space within Possible Low-Cost Dome Structures pg12 Figure 1.11 - Hoop and Meridian Directions pg13 Figure 1.12 ? Plaster Coatings & Shrinkage (Houben, Guillaud, 1994) pg14 Figures 1.13? Possible Internal Arrangements for Low-Cost Houses pg17 CHAPTER 2 Figure 2.1 ? The Basic Structural Elements pg19 Figure 2.2 ? Forces and Moments in Shell Structures pg19 Figure 2.3 ? Dome/Ring Equations? Parameters pg23 Figure 2.4 ? Sensitivity Analysis Dome Dimensions pg26 Figure 2.5 ? Graph of Meridian Forces - Sensitivity Analysis pg27 Figure 2.6 ? Graph of Hoop Forces - Sensitivity Analysis pg28 Figure 2.7 ? Graph of Meridian Moments - Sensitivity Analysis pg28 Figure 2.8 ? Effect of Varying the Shell Thickness - Sensitivity Analysis pg30 Figure 2.9 ? Graph of N? - Effect of Varying Ring Dimensions - Sensitivity Analysis pg31 Figure 2.10 ? Graph of M? - Effect of Varying Ring Dimensions - Sensitivity Analysis pg32 Figure 2.11 ? Y/L Ratio Parameters pg36 Figure 2.12 ? Fixed Base Reaction Forces pg37 Figure 2.13 ? The Ideal Shape of a Dome Structure pg38 Figure 2.14 ?The Catenary Equation Variables pg38 Figure 2.15 ? Graph of Maximum Hoop Force vs. Y/L ? Catenary pg39 Figure 2.16 ? Graph of Maximum Meridian Moments vs. Y/L ? Catenary pg39 Figure 2.17 ? Graph of Hoop Stresses ? Catenary Y = 4.92m pg40 Figure 2.18 ? Graph of Meridian Stresses ? Catenary Y = 4.92m pg40 v Figure 2.19 ? Graph of Maximum Hoop Force vs. Y/L ? Hemisphere pg43 Figure 2.20 ? Graph of Maximum Meridian Moments vs. Y/L ? Hemisphere pg43 Figure 2.21 ? Graph of Hoop Stresses ? Sectioned Hemisphere r = 3.5 pg43 Figure 2.22 ? Graph of Meridian Stresses ? Sectioned Hemisphere r = 3.5 pg44 Figure 2.23 ? Graph of Maximum Hoop Force vs. Y/L ? Parabola pg46 Figure 2.24 ? Graph of Maximum Meridian Moment vs. Y/L ? Parabola pg46 Figure 2.25 ? Graph of Hoop Stresses ? Parabola (y = 8m) pg47 Figure 2.26 ? Graph of Meridian Stresses ? Parabola (y = 8m) pg47 Figure 2.27 ? Graph of Maximum Hoop Force vs. Y/L ? Ellipse pg50 Figure 2.28 ? Maximum Meridian Moment vs. Y/L ? Ellipse pg50 Figure 2.29 ? Graph of Hoop Stresses ? Ellipse (B = 3.2m; Y = 3.2m) pg51 Figure 2.30 ? Graph of Meridian Stresses ? Ellipse (B = 3.2m; Y = 3.2m) pg51 Figure 2.31 ? Graph of N(?) ? Shells with 1m Cylinder Walls ? Structure Type B pg53 Figure 2.32 ? Graph of M(?) ? Shells with 1m Cylinder Walls ? Structure Type B pg53 Figure 2.33 ? Useable Space Parameters pg55 CHAPTER 3 Figure 3.1 ? Stress-strain behavior of FRC (Fulton 2001:253) pg61 Figure 3.2 ? Stress-Strain Curve Parameters pg66 Figure 3.3 ? Graph of Brickforce Yield Stress pg67 Figure 3.4 ? Graph of Wire Wrapping Yield Stress pg67 Figure 3.5 ? Graph of Chicken Mesh Yield Stress pg68 Figure 3.6 ? Line Diagram of the Inclined Plane Method pg70 Figure 3.7 ? Inclined Plane Friction Test Apparatus pg70 CHAPTER 4 Figure 4.1 ? Prototype 28m2 (Structure type (B)) pg75 Figure 4.2 ? Dimensions of the Final Structure (Prototype 28m2 Dome) pg76 Figure 4.3 ? 3D FEA Model (Final Structure) pg77 Figure 4.4 ? Wind Pressure on a Dome and a Cylinder (Billington, 1965:74) pg78 Figure 4.5 ? Partitioned Model (Wind Load- Patch load) pg80 Figure 4.6 ? Two Examples of Cracking Induced by Volume Changes pg81 Figure 4.7 ? Sections along Which Results are Presented pg83 Figure 4.8 ? Concentrated Load around a Skylight Opening (Billington, 1965:45) pg84 Figure 4.9 ? Graph of the Effect of Skylight Opening on Hoop Force pg85 Figure 4.10 ? Graph of the Effect of Skylight Opening on Meridian Moments pg85 vi Figure 4.11 ? Stress (Von Mises) around Supported and Unsupported Dome Openings pg86 Figure 4.12 ? Cracks Patterns around Openings at the Sparrow Aids Village pg87 Figure 4.13 ? Graph of Hoop Forces around Window Openings pg88 Figure 4.14 ? Graph of Meridian Moments around Window Openings pg89 Figure 4.15 ? Cracking on the inside face of a Window pg89 Figure 4.16 ? Graph of Hoop Force (1.2TL+0.9DL) pg90 Figure 4.17 ? Graph of Meridian Moment (1.2TL+0.9DL) pg91 Figure 4.18 ? Graph of Hoop Force (1.2TL+0.9DL ? Half Structure) pg92 Figure 4.19 ? Graph of Meridian Moment (1.2TL+0.9DL? Half Structure) pg92 Figure 4.20 ? Graph of Hoop Forces (ULS) ? Centre Section pg94 Figure 4.21 ? Graph of Meridian Moments (ULS) ? Centre Section pg94 Figure 4.22 ? Graph of Hoop Forces (ULS) ? Door Section pg95 Figure 4.23 ? Graph of Meridian Moments (ULS) ? Door Section pg95 Figure 4.24 ? Graph of Hoop Forces (ULS) ? Window Section pg96 Figure 4.25 ? Graph of Meridian Moments (ULS) ? Window Section pg96 Figure 4.26 ? Graph of Hoop Stresses ? Centre Section pg97 Figure 4.27 ? Graph of Meridian Stresses ? Centre Section pg98 Figure 4.28 ? Graph of Hoop Stresses ? Window Section pg99 Figure 4.29 ? Graph of Hoop Stresses ? Door Section pg99 Figure 4.30 ? Graph of Meridian Stresses ? Window Section pg100 Figure 4.31 ? Graph of Meridian Stresses ? Door Section pg101 Figure 4.32 ? Serviceability Deflections ? U2 pg102 Figure 4.33 ? Serviceability Deflections ? U1 pg102 CHAPTER 5 Figure 5.1 ? Stresses in a Masonry Wall (Curtin, 1985:60) pg105 Figure 5.2 ? Masonry Flexural Failure Planes (Curtin, 1985:61) pg107 Figure 5.3 ? Cracked Section Stress Block (Crofts & Lane, 2000:154) pg107 Figure 5.4 ? Design Stress Block, Compressive Stress pg108 Figure 5.5 ? Design Stress Blocks ? Compressive and Tensile Stress pg109 Figure 5.6 ? Design Stress Blocks - Tensile Stress pg110 Figure 5.7 ? HydraForm Splitter Block pg113 Figure 5.8 ? Graph of Hoop Moment & Resistance ? Centre Section (Dome) pg114 Figure 5.9 ? Graph of Hoop Stresses & Resistances ? Centre Section (Dome) pg115 Figure 5.10 ? Final Structure - Concrete Lintel pg116 Figure 5.11 ? Graph of Meridian Moment & Resistance ? Centre Section (Dome) pg116 vii Figure 5.12 ? Graph of Meridian Stresses & Resistances ? Centre Section (Dome) pg117 Figure 5.13 ? Wire Wrapping Detail pg118 Figure 5.14 ? Graph of Hoop Moments & Resistances ? Centre Section (Cylinder Wall) pg121 Figure 5.15 ? Graph of Hoop Stresses & Resistances ? Centre Section (Cylinder Wall) pg122 Figure 5.16 ? Graph of Meridian Moment & Resistance ? Centre Section (Cylinder Wall) pg122 Figure 5.17 ? Graph of Meridian Stresses & Resistances ? Centre Section (Cylinder Wall) pg123 Figure 5.18 ? Graph of Shear Stress in the Structure pg123 Figure 5.19 ? Foundation Detail (adapted from Dome Space drawing) pg125 CHAPTER 6 Figure 6.1 ? EcoShell Dome Construction Method (www.monolithic.com) pg127 Figure 6.2 ? Nubian Method of Dome Construction (Auroville Institute) pg128 Figure 6.3 ? Setting Out Equipment for the Foundation pg130 Figure 6.4 ? Setting Out of the Foundation pg130 Figure 6.5 ? Digging the Foundation pg131 Figure 6.6 ? Short Foundation Walls pg131 Figure 6.7 ? Placing of Foundation Reinforcing & Concrete pg132 Figure 6.8 ? Splitting the Splitter Blocks pg132 Figure 6.9 ? Construction of the Cylinder Wall pg133 Figure 6.10 ? The Inflatable Formwork pg134 Figure 6.11 ? Dome Construction pg135 Figure 6.12 ? Construction of the Arches pg136 Figure 6.13 ? Door Arch pg136 Figure 6.14 ? Un-plastered Dome pg137 Figure 6.15 ? Wire Wrapping around Door Opening pg137 Figure 6.16 ? Concrete Lintel pg138 Figure 6.17 ? The Completed 28m2 Dome pg138 Figure 6.18 ? Graph of the Strength of the Mortar pg140 Figure 6.19 ? The Schmidt Hammer pg141 Figure 6.20 ? Schmidt Hammer Results (Schmidt, 1950) pg142 viii APPENDIX B Figure B1 ? Alternative Structure ? Structure Type C pg154 Figure B2 ? Meridian Forces (Dome) ? Structure Type C pg155 Figure B3 ? Hoop Forces (Dome) - Structure Type C pg155 Figure B4 ? Meridian Moments (Dome) ? Structure Type C pg156 Figure B5 ? Meridian Forces (Cylinder) ? Structure Type C pg156 Figure B6 ? Hoop Forces (Cylinder) ? Structure Type C pg157 Figure B7 ? Meridian Moments (Cylinder) ? Structure Type C pg157 ix List of Tables CHAPTER 1 Table 1.1 ? Thermal Conductivity tests on Earth blocks (Lamb, 1998) pg16 CHAPTER 2 Table 2.1? Fixed Base Reaction Forces ? Catenary pg41 Table 2.2? Fixed Base Reaction Forces ? Sectioned Hemisphere pg44 Table 2.3 ? Parabola ? Fixed Base Reaction Forces pg48 Table 2.4? Fixed Base Reaction Forces ? Ellipse pg52 Table 2.5 ? Summary of the Best Shapes pg55 CHAPTER 3 Table 3.1 ? Basic Soil Requirements for CEB Production (Uzoegbo, 2003) pg58 Table 3.2 ? Properties of Cement Stabilized Earth Blocks pg58 Table 3.3 ? Mortar Compressive Strengths (SABS 0164: Part 1 Table 1) pg61 Table 3.4 ? Compressive Strength Tests Results pg62 Table 3.5 ? Tension Test Results ? Fibre Plaster pg64 Table 3.6 - Concrete Block Sliding Down Mortar Surface pg72 Table 3.7 - One Layer of DPC Wrapped around Concrete Block pg73 Table 3.8 - Two Layers of DPC Wrapped around Concrete Block and Mortar Surface pg74 CHAPTER 4 Table 4.1 ? Load Combinations According to SABS 0160 ? 1989 pg83 CHAPTER 5 Table 5.1 ? Wire Wrapping Calculation ? Door Section pg119 Table 5.2 ? Wire Wrapping Calculation ? Window Section pg119 Table 5.3 ? Foundation Loading pg124 CHAPTER 6 Table 6.1 ? Requirements for Mortar (SABS 0164:1 Table 1) pg139 Table 6.2 ? Mix Proportions for Mortar (SABS 0249 Table 5) pg139 Table 6.3 ? Foundation and Floor Slab Compressive Strengths pg142 Table 6.4 ? Cost Comparisons of Low-Cost Housing Schemes pg144 x APPENDIX A Table A1 ? 1.5DL ? Centre Section Results pg152 Table A2 ? 0.9DL+1.3WL (Suction side) ? Centre Section Results pg153 xi List of Symbols DESCRIPTION SYMBOL Radius of the dome a Cross-sectional area A Area of Steel As Height of the ring beam b Compressive Force C External Pressure Coefficient Cp Ring compression induced @ skylight opening C?0 Thickness of the ring beam d Horizontal displacement of the dome @ the base D10D Horizontal displacement of the ring beam D10R Rotation of the dome @ the base D20D Rotation of the ring beam D20R Compatibility Matrix Variable D11 Compatibility Matrix Variable D12 Compatibility Matrix Variable D21 Compatibility Matrix Variable D22 Dead Load DL Young?s Modulus E Characteristic compressive strength fk Yield Strength of Steel fy Force F Design vertical load ga Design vertical stress gd Shell Thickness h Horizontal thrust @ base of dome H Horizontal thrust caused by skylight opening H?0 Moment of inertia I Constant depending on altitude kp Base diameter of the dome L Live Load LL Moment M Allowable moment Mallow xii DESCRIPTION SYMBOL Applied moment Mappl Moment @ base of dome M? Hoop moment M? Meridian moment M? Hoop force N? Meridian force N? Hoop membrane force N?? Meridian membrane force N?? Free stream velocity p Wind pressure pz Point load @ skylight opening P Uniformly distributed load q Radius of ring beam r Thickness of fibre plaster t Tensile force T Characteristic wind speed vz Volume V Weight W Wind Load WL Cartesian x ? coordinate x Cartesian y ? coordinate y Half of the ring beam height y0 Height of the dome Y Cartesian z ? coordinate z Capacity reduction factor ? Horizontal displacement @ base of dome ?H Rotation @ base of dome ?? Strain ? Angle in the vertical plane of the dome ? Material safety factor ?m Friction coefficient ? Stress ? Poisson?s ratio ? 1 1. Introduction 1.1 Statement of the Problem There is a large demand for low cost housing in South Africa. The 2001 South African Census showed that approximately 1.9 million households are informal and approximately 1.7 million are classified as traditional dwellings (Figure 1.1). Figure 1.1 ? Distribution of South African Houses (RSA Census, 2001) The people living in traditional houses live mainly in the rural areas of South Africa. These houses could be improved on, making them more durable. The number of informal houses is a concern, as a large proportion of these houses are located in squatter camps (shanty towns) in urban areas. The living conditions in these areas are poor. There is a great need for a more formal type of housing that would provide improved thermal properties during winter, as well as a greater resistance to the elements. Current mass housing solutions focus on conventional methods of construction (i.e. rectangular brick structures with roof trusses). 2 Innovative and cost effective materials and construction procedures are needed in order to improve the growing housing shortage. Dome structures have been used effectively, in studies done at The University of the Witwatersrand, as an alternative housing form. This dissertation seeks to investigate the dome structure as a low cost housing alternative. The problem of cost effective materials is addressed by using cement stabilized earth blocks (CEB?s) in the construction of the dome. 1.2 Aims of the Investigation The main objectives of this dissertation are: square4 To identify the optimal shaped dome structure that can be used in low cost housing square4 To investigate the important design criteria with regard to domes square4 To utilize affordable materials (compressed earth blocks) that can be acquired in remote areas square4 To investigate different methods of dome construction square4 To construct a durable, architecturally and structurally efficient low cost dome house 1.3 Method of Investigation The method of the investigation is shown in Figure 1.2 (pg. 4). This investigation was design orientated and therefore the solution procedure was iterative. Many different shapes can be used in the design of shell structures, and therefore the most important step in any shell analysis is determining the best shape for the structures given function. A shape investigation was undertaken in order to determine the optimum shape of the structure. In order to perform this investigation the material properties needed to be defined. This was done through a literature investigation. Once an optimum shape was found the design of the structural elements (masonry, ring 3 beam) was performed. Through investigation it was found that a masonry dome solution for low-cost housing already existed (Dome Space solution). The shape of this dome was optimal from an architectural point of view (i.e. maximum useable space). However, specific problems were identified with this solution (e.g. lack of ductility allowing thermal cracking and cracking induced in regions of high stress), and therefore some engineering of the solution was required. The construction technique used to build the dome was well documented. This was done in order to check the quality of the materials on site, and to compare the construction procedure used to traditional dome construction techniques (e.g. Nubian method of construction used by Magaia (2003)). 4 Figure 1.2 ? Flow Diagram of the Investigation 5 1.4 A Brief History of the Dome Domes have been used throughout the ages as a housing form, or an element of a housing form (roof structure). African and aboriginal societies built domes by planting branches in the ground and weaving the dome shape (Kirchner, 1988). Figure 1.3 shows a typical Zulu hut in Kwazulu-Natal, South Africa. Figure 1.3 - Zulu Hut (KwaZulu-Natal) The dome was also used by the Musgum tribe of Cameroon (Gardi, 1973). This parabolic dome consists of a highly cohesive earth shell, 15-20cm (5.9- 7.9 inches) thick at its base, 5cm (1.97 inches) thick at the top and 7-8m (23-26 ft) high. Figure 1.4 shows a typical Musgum dome. Figure 1.4 - Musgum Farmstead 6 The dome shape has also been used historically to span large distances, as reinforced concrete and steel have been relatively recent developments on the historical scale. The Moguls, Egyptians, Byzantines and Romans used domes and vaults extensively. Examples of these are The Shrine of the Living King (Samarkand) built by the Moguls, The Pantheon in Rome and The Temple of Ramses II (near Aswan). The Temple of Ramses II was constructed out of earth bricks in 1290BC and parts of the structure, such as the vaults where the priests stored grain, are still standing (Melaragno, 1991). This illustrates the great potential of earth as a durable construction material. Figure 1.5 - The Ramesseum Storage Vaults, Gourna, Egypt . 7 1.5 Recent Developments in Compressed Earth & Dome Construction This section introduces some of the work that has been done in the field of earth and dome construction. The first two reports were done at The University of the Witwatersrand, and they provided some of the parameters used in this dissertation. 1.5.1 Fibre Reinforced Soil Crete Blocks for the Construction of Low-Cost Housing ? Rodrigo Fernandez (University of the Witwatersrand, 2003) This investigation focused on the production of fibre reinforced cement stabilized blocks. The objective was to study different types of soil, curing processes and reinforcements, in order to identify the most important parameters in the production of strong, durable blocks. The project was undertaken by the University of the Witwatersrand and the Lausanne Federal Institute of Technology (EPFL) (Fernandez, 2003). This report was useful to this project as it provided design information about the HydraForm earth block. Tests were performed to determine dry compressive strength, flexural strength, tensile strength and the modulus of elasticity of the blocks. Scanning electron microscope analysis was also used in this investigation. It was found that the nature of the soil, particularly the clay content and plasticity index were important. These variables could be optimized to improve the mechanical properties of the blocks. It is important to note that fibres did not always contribute to strength, and in some cases actually reduced the strength of the blocks (Fernandez, 2003). 8 1.5.2 Domes in Low-Cost Housing ? S.J. Magaia (University of the Witwatersrand, September 2003) This investigation involved the construction of a dome shaped low-cost house in a rural area close to Maputo, Mozambique. Figure 1.6 shows the final product of this investigation. The study sought to prove the viability of this type of structure for the use of low-cost housing. Furthermore, the study was aimed at achieving a self-construction solution, using local materials to construct the dome. The concept was proved to be viable (Magaia, 2003). However, architectural issues were not adequately considered. Wasted space around the perimeter of the dome, lack of internal light and the unpleasant appearance of the dome were issues that were not sufficiently explored. Figure 1.6 - Low-Cost Dome built by S.J. Magaia(2003) 9 1.5.3 The Monolithic Dome Solution Figure 1.7 ? EcoShells in Haiti. (www.monolithic.com) The EcoShell is a dome home designed by Monolithic Dome of Texas. The above figure shows an EcoShell constructed in Haiti as a form of low-cost housing. The dome is constructed by spraying concrete onto an inflated formwork that has reinforcing bars attached to it. The EcoShell is hemispherical in shape. The benefit of this type of construction is the speed at which the structure can be erected (Garrison, 2004). However, the materials and equipment needed to construct sprayed concrete domes are relatively complex and costly. In a rural environment this type of construction may be difficult to implement. 1.5.4 The Dome Space Solution Figure 1.8 ? Dome Homes built by Dome Space 10 Figure 1.8 shows two masonry domes built by Dome Space, South Africa. The domes are hemispherical in shape and were built using inflatable formworks. The dome on the left was built for the Sparrow Aids Village and the dome on the right for a village in the same area. Extensive cracking was observed on the Sparrow Aids Village Dome. The cause of cracking in brick domes was a primary concern in this investigation, and is discussed further in sections 4.4.2 and 4.4.3. 1.5.5 The Tlholego Eco-Village The Thholego Eco-Village is situated in the North West Province of South Africa. A masonry dome was built above a cylinder wall (to eaves height), and the structure was used as a kitchen for the local community. Cracking was also observed. Poor waterproofing, shrinkage and thermal cracking was postulated as the cause of cracking. Figure 1.9 shows the Tlholego Eco-Village Dome kitchen. Figure 1.9 ? Dome Kitchen ? Tlholego Eco-Village 1.6 Architectural Considerations This section deals with a few key architectural issues which need to be addressed when building any structure. The issues discussed can be summarized under the following headings: ? Shape ? Structural Strength ? Rain Penetration & Damp Proofing 11 ? Thermal Performance ? Lighting ? Internal Arrangement of Walls ? Overall Aesthetics 1.6.1 Shape The shape of the dome structure is perhaps the most important variable that needs to be investigated when designing a brick dome. Three types of dome structure were investigated (see figure 1.10). These were: square4 A dome structure from ground level - Type (A) square4 A dome placed on a short cylinder wall - Type (B) square4 A section through a dome placed on a vertical cylinder wall (with a ring beam) - Type (C) In this dissertation emphasis is placed on attaining the most efficiently shaped dome. The efficiency of the structure was measured in two ways: First, the shape and second, useable space. The dome curves in two directions making it very difficult to fit furniture into the home. Floor space is wasted when walls slope to the base of the structure. Figure 1.10 shows the three types of domes investigated in this report, as well as the concept of useable space. The units are given in metric millimeters. 12 Figure 1.10 ? Useable Space within Possible Low-Cost Dome Structures In order to improve the amount of useable space within a dome the diameter of the dome needs to be increased so that items can be placed closer to the wall (structure type A). By doing this the floor area is increased, but space is still wasted around the edges of the dome. The two alternative shapes (B & C) in figure 1.10 incorporate a straight cylinder wall. Structure (B) comprises of a full dome roof and a short cylinder wall 1m high. Structure (C) comprises of a sectioned dome roof with a concrete ring beam resting on a cylinder wall to door height. Figure 1.10 shows that structures (B) and (C) are more desirable than structure (A) from a useable space point of view. 1 foot = 304.8mm 13 1.6.2 Structural Strength The dome element of the structure was chosen for its structural strength as well as the savings envisaged by replacing a conventional roofing system with a monolithic element. The dome is defined as a doubly curved surface element. The dome?s doubly curved surface allows it to carry loads (especially its dead weight) very effectively. The load is carried primarily by membrane action. Moments and shears are limited to the area around the base (or boundary) of the shell. Stiff horizontal rings around the shell limit the deformation in the meridian direction. Figure 1.11 shows the hoop and meridian directions in a shell. Figure 1.11 - Hoop and Meridian Directions (Magaia, 2003) 1.6.3 Rain Penetration & Damp Proofing It is undesirable and unhealthy to have moisture within a living environment. Rain penetration can be prevented by careful detailing of the structure. The entire structure was plastered in order to prevent rain penetration. It is important to plaster the structure in stages in order to limit shrinkage. It is recommended (Doat, 1985) that plaster is applied in three coats to a vertical wall. It is very important that strong cement plasters are avoided. These form a 14 rigid surface above earth walls, and since the earth wall is not as rigid as the plaster the plaster tends to crack. Temperature and moisture cycles can cause further cracking. Moisture enters the wall and erodes parts of the wall away (Magaia, 2003). A possible solution to this is using fibre reinforcement or wire mesh in the plaster (William-Ellis, 1947). These methods of reinforcing the plaster are explored to a greater extent in the materials investigation section. Figure 1.12 shows the shrinkage cycle of plaster on a vertical wall. For dome structures, the dome is very stiff owing to its shape. It is postulated that the stiffness of the shell prevents the contraction of the plaster (during shrinkage), causing the plaster to crack. Figure 1.12 ? Plaster Coatings & Shrinkage (Houben, Guillaud, 1994) 15 A reduction in the cement content in the plaster mix will help to reduce the stiffness of the plaster surface, and a reduction in the water content will limit the shrinkage of the plaster. It is important to note that plastering should take place on moderately warm and slightly humid days. Ten to twenty meter squared sections should be plastered at one time. All walls that are started should be finished on the same day, and the plastering should not extend to ground level due to capillary action which could cause moisture ingress. 1.6.4 Thermal Performance The thermal performance of domed shaped houses is very favorable (Kirchner, 1988). Convection currents occur in well ventilated domes allowing more even temperatures during the summer months. The final structure chosen for construction had three windows and a door, at quarter points. There were no openings in the roof (except for very small vents in the skylight). The materials used to build the dome also contributed to the favorable thermal performance of the structure. A good building material is one with a high heat capacity (it can store a fair amount of heat) and a low thermal conductivity (it retains the heat it stores). Thermal conductivity and heat capacity tests have been done on compressed earth blocks at The University of the Witwatersrand by Lamb (1998). The tests were performed on three types of earth blocks and a standard clay brick in order to compare their thermal conductivity and their heat capacity. Table 1.1, overleaf, summarizes the relevant findings of Lambs? investigation. 16 TYPE OF BLOCK THERMAL CONDUCTIVITY (K) [W/M?C] HEAT CAPACITY [KJ/KG?C] DENSITY [KG/M3] HydraForm Earth block ? Quaternary Sand (14% clay) ? 6% Cement 0.578 0.853 1 780 (111 lb/ft3) Clay Fired Brick ? Standard according to SABS 1215 0.82 0.8 1 826 (114 lb/ft3) Table 1.1 ? Thermal Conductivity tests on Earth blocks (Lamb, 1998) From table 1.1 we can see that the HydraForm Earth block1 (used in this dissertation) has a lower thermal conductivity than the clay fired brick, which means it will retain more heat. It also has a greater heat capacity which means it will absorb more heat than clay fired bricks. 1.6.5 Lighting The dome that was constructed was limited to three windows, a central skylight and a door. The skylight was positioned at the center of the dome as this is the most structurally efficient place to put it. It provides a central core of light within the house. A complaint with regard to low-cost domes built in the past was that they were very dark inside. The central skylight illuminates the inside of the structure and improves the lighting problem. 1.6.6 Internal Arrangement This aspect of the design is of great importance and an architect may be needed to maximize the efficiency of the space. There are a large proportion of houses in both rural and urban areas that have four or more people living in one household (RSA Census 2001). This poses challenges to designers of low-cost houses as they need to be designed as small as possible and accommodate as 1 HydraForm are a South African company who specialize in the production of cement stabilized earth blocks. 17 many people as possible. Figure 1.13 shows possible internal arrangements of low-cost dome houses. The arrangement will be dictated by the tenants? affluence. These internal arrangements were designed to accommodate between three to six people. Pit latrines on the outside of the house can be used to save space within the structure. Internal plumbing could be utilized with a service core supplying water to a basic kitchen and bathroom area. However, this solution would increase the costs. Figure 1.13 also shows an internal arrangement for a more affluent family (on the right). Cupboards and packing space can be installed around the perimeter of the house to save space. Figures 1.13? Possible Internal Arrangements for Low-Cost Houses 18 2. Shape Investigation 2.1 Methods of Shape Investigation There are three main methods of designing shell structures. The first two involve the analysis of predetermined geometric shapes. These are classical thin shell theory and finite elements. The third method is an empirical method; this method involves scaled models of the structure. The Swiss engineer, Heinz Isler, used models to find funicular shapes by suspending materials so that they hung in pure tension. These models were then frozen and inverted, and a shape in perfect compression resulted. In this investigation classical shell theory and finite element analysis (FEA) are used to determine the best structural shapes. Three types of shape are presented: square4 A dome structure from ground level - (Type A) square4 A dome placed on a short cylinder wall - (Type B) square4 A section through a dome placed on a vertical cylinder wall (with a ring beam) - (Type C) 2.2 Classical Thin Shell Theory 2.2.1 The Structural Elements Figure 2.1 shows the basic structural elements of a dome home. The ring beam is an integral part of any dome as it prevents the dome from kicking out. It can be placed at ground level as a foundation or on a wall as shown in figure 2.1, overleaf. 19 Figure 2.1 ? The Basic Structural Elements 2.2.2 Forces and Moments in the Structure This report assumes that the reader has a basic understanding of shell theory and therefore the equations have not been proven from first principles. The forces and moments that exist in a dome structure are summarized in figure 2.2. Figure 2.2 ? Forces and Moments in Shell Structures Shear Force Q(y) 20 2.2.3 Classical Dome Theory Classical dome theory was used to check the accuracy of the FEA analysis in the sensitivity analysis part of this chapter. This theory is well documented by Billington (1982). The solution procedure involves solving for the membrane forces and deformations without any restraints (supports at the edges) on the structure, and then correcting these results by applying a support restraint. Two sets of equations exist in this solution. They are the membrane equations and the boundary equations. According to Billington (1982), the solution can be broken up into four parts. The first part is called primary system. This system is based on membrane theory. The equations for the membrane forces with a uniformly distributed load (self weight load) applied to the shell surface are presented below. )cos1(' ?? + ? = aqN (2.1) )cos cos1 1(' ??? ?+= aqN (2.2) where, a is the radius of a spherical dome q is the uniformly distributed load ? is the angle measured from the apex to the base of the dome The second part of the solution involves calculating the errors. The errors are the deformations of the dome according to membrane theory. The equations presented are for a UDL load: 10 2 sin)cos cos1 1( D Eh qa H =? + + =? ??? ? (2.3) 21 20sin)2( DEh aq =+?=? ??? (2.4) where, 10D is equal to the horizontal membrane deformation 20D is equal to the membrane rotation E is equal to Young?s Modulus h is equal to the shell thickness ? is equal to Poisson?s ratio The third part of the solution is to solve for the corrections. Eh HaHDH ?? 22 11 sin2 ==? (2.5) Eh Ma MDH ?? ? ? sin2 22 12 ==? (2.6) Eh HaHD ??? sin2 22 21 ==? (2.7) Eh Ma MD ??? ? 23 22 4 ==? (2.8) where, H and M? are the shear forces and moments respectively, applied at the base (edge) of the shell structure in order to correct the membrane displacements. The fourth part of the solution is to determine H and M?, using compatibility equations. The compatibility equations are 0121110 =++=?? ?MDHDDH (2.9) 0222120 =++=?? ?? MDHDD (2.10) 22 The final steps in the solution are to calculate the shear forces, deformations and moments in the shell due to boundary effects. These equations can be seen in Billington (1982). The membrane forces are combined with the boundary effects. 2.2.4 Classical Dome/Ring Theory Dome/Ring theory is the theory applied to dome structures with a ring beam at the base of the structure. The structure does not behave as if it were fixed or pinned, but somewhere between these two conditions. The ring beam will allow a certain amount of rotation and outward displacement at the base, which a fixed base would not. Therefore, two new errors are introduced and these are added to the dome membrane errors to obtain the total dome/ring errors. Ebd Nr d ey D R ?? ')12(cos 2 2 0 10 += (2.11) R EI erN D R 2 20 '?? = (2.12) RD DDD 101010 += (2.13) RD DDD 202020 += (2.14) The variables in the above equations are described in figure 2.3, overleaf. 23 Figure 2.3 ? Dome/Ring Equations? Parameters The additional correction equations are: Ebd Hr d y HDRH 2 2 2 0 11 ) 121( +==? (2.15) 3 0 2 21 12 Ebd Hyr HD R ? ==?? (2.16) 3 0 2 12 12 Ebd Myr MDRH ? ? ? ==? (2.17) 3 2 22 12 Ebd Mr MDR ??? ==? (2.18) After this step the compatibility equations are compiled using equations 2.5 ? 2.8 and 2.15 ? 2.18. The rest of the equations are identical to the classical dome theory solution. 2.3 Basic Finite Element Analysis (FEA) Theory Finite element analysis is used as an approximate solution to engineering problems. It is important to understand the limitations of this type of analysis, as well as the methods of assessing and improving the analysis when modeling 24 a structure. In this section a few practical issues regarding finite elements are discussed. The equations for the finite elements used are not presented. 2.3.1 Types of Elements Used There are many shell elements that can be used to analyze a dome structure. This is due to the fact that shell elements are not fully compatible (the displacements are not always continuous along plate boundaries). Therefore errors can result in the analysis. For this reason a sensitivity analysis was done in order to check the accuracy of the FEA analysis against dome and dome/ring theory. The final structure modeled in this report used three dimensional shell elements. The shape investigation used axi-symmetric finite elements owing to the symmetry of the problem. NAFEMS (National Agency for Finite Element Methods and Standards) suggests a few guidelines when choosing a shell element (Blitenthal, 2004). These are: ? Quadratic element types are more exact than linear elements. ? Linear elements are stiffer and produce lower displacements and stresses. ? Quadratic elements should be used for curved problems as they produce a better approximation. Linear elements will cause stress discontinuities along shell element boundaries. ? Shell elements are not accurate where there is a sharp change in geometry. This was one of the main concerns with the FEA analysis as there is a join between the ring beam and the dome structure. The sensitivity analysis proved that this was not a problem. 25 2.3.2 Methods of Improving FEA Accuracy There are a few methods of improving the accuracy of the FEA analysis. The first is to use higher order elements (e.g. quadratic instead of linear). The majority of the other techniques involve the meshing of the model. These include: ? Using a structured mesh. A structured mesh comprises of square elements placed in a regular pattern. This type of mesh is not possible when modeling doubly curved surfaces. In this case, a triangular mesh, with triangles that are as close to equilateral as possible, or an irregular quadrilateral mesh, with quadrilaterals as close to square as possible, can be used. ? Using the mesh checking facilities provided by the FEA program to check aspect ratios (no greater than 3), free edges, angular distortion and internal element angles. ? Using second order elements if an automatic mesh generator is used. Once the analysis has been completed the results should be evaluated and if necessary the mesh should be refined and the model reanalyzed. As mentioned earlier, a sensitivity analysis is important in checking the accuracy of the FEA model and can be used to find a suitable mesh. 2.4 Sensitivity Analysis 2.4.1 Description of the Analysis It is extremely important when using finite element analysis to check whether the model is yielding accurate results. A sensitivity analysis was undertaken before the shape investigation in order to check the accuracy of the different types of finite elements that can be used to model the structure in AbaqusTM. Two types of elements were investigated. These were the full 3D rotational shell element and the 2D axi-symmetric deformable shell element. Both 26 elements use a quadratic function to model their deflections. The results from these analyses were compared with a traditional shell analysis of the same structure. The sensitivity analysis was also used to check the influence of certain parameters on the results of the FEA analysis. Fixity at the base of the structure was investigated, as well as the effect of changing the thickness of the dome (stiffness) and changing the applied load on the structure. A section through a hemisphere, with a radius of 3m (9.84 ft) and a height of 2.5m (8.2 ft) was used for this analysis. Three analyses were performed. In first analysis a fixed base was used, in the second a ring beam base was used (depth = 0.275m (11.2 inches), width = 0.29m (11.4 inches)), and in the third the dome was pinned. The properties of 30 MPa (4 351 psi) concrete were used in this analysis. A uniformly distributed load (UDL) of 4.69 kN/ m2 (0.68 psi) was applied to the structure. This load included the dead load (self weight of the structure), as well as a live load of 0.5kN/ m2 (0.07 psi). The following figure shows the dome used for the sensitivity analysis: Figure 2.4 ? Sensitivity Analysis Dome Dimensions The results obtained from the three different analyses are presented below. They include the Meridian Force, the Hoop Force and the Meridian Moments. The Hoop Moments were excluded as they can be calculated by multiplying the meridian moments by Poisson?s ratio. The accuracy of the finite element 27 analysis depends on the size of the elements chosen and the type of element chosen. Therefore, a fine mesh and elements with mid-side nodes were used in order to achieve good results. 2.4.2 Sensitivity Analysis Results The results are presented along horizontal (x-direction) axis of the structure, from the centre of the dome to the dome base. Figure 2.5 ? Graph of Meridian Forces - Sensitivity Analysis Meridian Forces (Sensitivity Check) -14.0 -12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Distance from Centre of Section (m) Fo rc e [k N/ m ] Classical Shell Theory (Fixed) 1D-Finite Element (Fixed) 3D-Finite Element (Fixed) Classical Shell Theory (Ring Beam) 1D Finite Element (Ring) Classical Shell Theory (Pinned) TOP OF DOME DOME BASE 1m = 3.28 ft; 1kN/m = 0.06854 kips/ft 28 Figure 2.6 ? Graph of Hoop Forces - Sensitivity Analysis Figure 2.7 ? Graph of Meridian Moments - Sensitivity Analysis Hoop Forces (Sensitivity Check) -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Distance from Centre of Section (m) Fo rc e [k N /m ] Classical Shell Theory (Fixed) 1D-Finite Element Fixed 3D-Finite Element (Fixed) Classical Shell Theory (Ring Beam) 1D-Finite Element (Ring) Classical Shell Theory (Pinned) TOP OF DOME DOME BASE 1m = 3.28 ft; 1kN/m = 0.06854 kips/ft Meridian Moments (Sensitivity Check) -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Distance from Centre of Section (m) M o m e n t [k N m /m ] Classical Shell Theory (Fixed) 1D-Finite Element (Fixed) 3D-Finite Element (Fixed) Classical Shell Theory (Ring Beam) 1D-Finite Element (Ring) Classical Shell Theory (Pinned) TOP OF DOME DOME BASE 1m = 3.28 ft; 1kNm/m = 0.2248 ft k/ft 29 It can be seen from the above graphs that the AbaqusTM results follow classical shell theory results quite closely. It can also be noted that the 2D (axi- symmetric) method of analysis yielded results almost exactly the same as the shell theory results. This simplified the shape investigation as a 2D analysis could be done instead of a full 3D analysis. The introduction of a ring beam at the base of the shell increased the hoop forces in the structure. The ring beam also affected the meridian moments, as seen in figure 2.7. In the ring beam analysis, the moments follow the same trend as the pinned base moments in the upper section of the dome. However, at the base of the dome the moments are closer to the values of the fixed base analysis. 2.4.3 Effect of Varying the Stiffness of the Structure This section discusses the relationship between shell thickness (stiffness) and the forces in the dome, as well as the effect of varying the ring beam dimensions (stiffness at the base of the structure). This information is useful to determine a reasonably sized ring beam for the shape investigation. Varying the Shell Thickness The results presented are for a dome surface analyzed using shell theory. The self weight of the dome was increased according to its thickness. The graph overleaf shows the relationship between the shell thickness and the forces acting in the structure. 30 Figure 2.8 ? Effect of Varying the Shell Thickness - Sensitivity Analysis The relationship between the shell thickness and the forces is linear, whereas the relationship between the shell thickness and the moments is a polynomial of the 3rd order. It is therefore very important to select the thinnest possible shell section in order to minimize the forces and moments in the structure ? stiffness attracts force. A thinner section will dissipate the forces quicker than a thicker sectioned thin shell. The moment region at the base of a thick shell (e.g. a concrete thin shell) is larger than the moment region at the base of a thinner shell (e.g. a thin steel shell). Young?s Modulus (E) has no effect on the forces and moments as long as the same material is used throughout the shell. However, it does have an effect on the calculated deformations. Effect of varying shell thickness R2 = 1 R2 = 1 -70 -60 -50 -40 -30 -20 -10 0 10 20 0 0.1 0.2 0.3 0.4 0.5 Shell thickness (m) (Fo rc e s a n d M o m en ts ) /( Fo rc es an d M o m en ts th k = 0. 05 m ) Hoop Forces Meridian Forces Meridian Moments Linear (Hoop Forces) Poly. (Meridian Moments) 1m = 3.28 ft 31 Varying the Ring Beam Dimensions The following graphs show that as the ring dimensions are increased the forces and moments in the structure tend towards the fixed support case. It can also be seen that by including a ring beam the hoop forces in the structure are increased considerably. These findings prove the basic structural concept that stiffness attracts load. An infinitely stiff ring beam will attract the maximum amount of load (fixed base). If a smaller sized ring beam is used more load must be carried by the shell. Thus, the hoop forces in the shell are greater. N? - Hoop Forces - Effect of Ring Beam -40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Distance Fron Centre of Section Fo rc e [kN /m ] Ring @ Base a=500mm ; d=300mm Ring @ Base a=1000mm ; d=300mm Fixed Base Ring @ Base a=1000000mm ; d=300mm TOP OF DOME DOME BASE Figure 2.9 ? Graph of N? - Effect of Varying Ring Dimensions - Sensitivity Analysis 1m = 3.28 ft; 1kN/m = 0.06854 kips/ft 32 M? - Meridian Momentss - Effect of Ring Beam -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Distance From Centre of Section M o m en t [ kN m /m ] Ring @ Base a=500mm ; d=300mm Ring @ Base a=1000mm ; d=300mm Fixed Base Ring @ Base a=1000000mm ; d=300mm TOP OF DOME DOME BASE Figure 2.10 ? Graph of M? - Effect of Varying Ring Dimensions - Sensitivity Analysis 2.4.4 Summary of Sensitivity Analysis Findings The findings of the sensitivity analysis and their effect on the shape investigation were: square4 An axi-symmetric finite element analysis could be used in the shape investigation, reducing computing and modeling time. square4 The shell thickness and load on the shells were kept constant in the shape investigation in order to compare the forces in the different shapes. square4 The hoop forces and meridian moments were found to be the main contributors to the tension stresses in the structure and therefore needed to be minimized. A fixed base analysis was performed in order to obtain accurate kicking out forces2 at the bases of the dome shapes modeled from ground level. 2 The kicking out force is the horizontal component of the meridian membrane force. This force acts at the base of the dome and pushes the dome support outward. 1m = 3.28 ft; 1kNm/m = 0.2248 kips kips/ft 33 square4 A ring beam needed to be incorporated into the shape investigation, as the forces in the dome when a ring beam is used are somewhere between the fixed and pinned base forces. Therefore a reasonably sized ring beam was chosen in the shape investigation and the results of the analysis with a ring beam were compared to the fixed and pinned base results. 2.5 Shape Analysis 2.5.1 Description of the Analysis This investigation was undertaken in order to obtain the most structurally efficient shape for the proposed low cost house. Three types of structure were investigated in the shape analysis. These were: square4 Type (A) - A dome structure from ground level square4 Type (B) - A dome built directly onto a short cylinder wall square4 Type (C) - A section through a dome placed on a vertical cylinder wall (with a ring beam) The shapes used in the above three structures were catenaries, parabolas, hemispheres and ellipses. The maximum height of the overall structure was limited to approximately four meters for building purposes. This constraint limited the shapes that could be used for the roof structure in option (C), above. The shapes were modeled in AbaqusTM using axi-symmetric finite element analysis. The material properties of the HydraForm (a South African earth brick manufacturer) earth blocks are given below: ? Density = 1 950kg/m3 (121.7 lb/ft3) ? Young?s modulus = 3 500 MPa (507.6 ksi) ? Poisson?s ratio = 0.2 34 ? Shell thickness = 0.14 m (5.51 in.) A standard ring beam was used to check the effect it has on the forces and moments in the structure. The dimensions of the ring beam for dome types (A) & (C) were assumed to be: square4 Width (b) = 0.29m (11.4 in.) square4 Depth (d) = 0.275m (10.8 in.) A body force of 28, 7 kN/m3 (0.105 lbf/in3) was applied to the different shapes in this investigation. A body force was used as this the standard vertical load that can be used in AbaqusTM (the FEA program). Using the same force enabled a direct comparison of the different shapes. The force was determined as follows: Brick Density x Gravitational Constant 33 33 /13.191000/19130 /1913081.9/1950 mkNmN mNmkg =? =? This load was then factored using the dead load factor of 1.5. 1.5 Dead Load 3/7.2813.195.1 mkN=? (0.105 lbf/in3) The load factor of 1.5 DL was used in accordance with SABS 0160: Part1 (1989). This load factor yielded the maximum moments and forces within the final structure. It is important to note that the shape investigation is a comparative investigation and any reasonable load can be used to compare the effectiveness of the shapes.3 3 The linear relationship between the applied force, and the resulting forces and moments in a dome structure can be seen in Figure 2.9. 35 2.5.2 Shape Evaluation Criteria The important criteria for the evaluation of the different shapes can be summarized into four questions: ? Which shape yields the smallest tension forces (hoop forces) and moments? ? Where are the tension regions in the structure? ? Which dome shape produces the smallest kicking out forces? ? Which shape is the most suitable for low-cost housing (cost, useable space)? The first three questions deal with tension in the structure. Tension stresses need to be minimized as they cannot be adequately resisted by the masonry and therefore, more expensive materials (e.g. reinforcing bars) are required to resist the stresses. The fourth question concerns the aesthetics and constructability of the structure. 2.5.3 Presentation of the Analysis Results The results for each shape are summarized into three sections. ? The first section shows the maximum and minimum hoop forces and meridian moments in the different shapes. These forces are plotted against a Y/L (height/base diameter) ratio. The Y/L ratio is the ratio of the height of the structure to the diameter (figure 2.11). Moments are shown positive clockwise and positive forces are tensile. 36 Figure 2.11 ? Y/L Ratio Parameters ? In the second section, the stresses are plotted against the x-distance shown in figure 2.11. Basic elastic stress formulae are used to calculate the stresses. This formula is shown below: (2.19) where: F = force; N(?) ? hoop force ; N(?) ? meridian force M = momen; M(?) ? hoop moment ; M(?) ? meridian moment A = area [m2] Z = section modulus [m3] The elastic stress formula is applied to the hoop and meridian directions to check which regions of the best performing structures are in tension, and which faces (inside or outside) the tension stress is acting. It is important to note that positive values (in the stress plots) denote tension while negative values denote compression. ? The third section presents the kicking out forces at the base of the shape. The kicking out force is the horizontal component of the meridian membrane force. This force acts at the base of the dome and Z M A F ?=? 37 pushes the dome support outward. This parameter is critical when designing domes supported at their bases with ring beams or walls (structure type C). The magnitude of the force, that pushes the dome outwards (RF1), determines the quantity of reinforcing in the ring beam. This force creates tension at the base of the structure, thus the need for reinforcing. In order to improve economy of the structure this force must be minimized. Figure 2.12 shows the AbaqusTM sign convention of the reaction forces at the base of the dome. Figure 2.12 ? Fixed Base Reaction Forces 2.6 Shape Analysis Results (Structure Types A & C) 2.6.1 The Catenary A catenary shape is obtained when a chain is held at two ends and left to hang freely. This shape is in perfect tension when the only force acting on it is its self weight. If this shape is inverted it produces a structure in perfect compression. A catenary is the perfect shape for an arch or barrel vault structure. However, for a dome structure, experimental techniques are needed to find the optimum shape. Figure 2.13 illustrates why a catenary is the optimum shape for a barrel vault but not a dome. 38 )cosh( a x ay = Figure 2.13 ? The Ideal Shape of a Dome Structure The equation of a catenary is: (2.20) Y = Height of shape [m] Figure 2.14 ?The Catenary Equation Variables The base diameter of the catenary was set at 6.4m (21 ft) in this investigation (the diameter of a reasonably sized low cost home). The y-value in the equation above is the variable that was changed in order to generate the different shapes. The following sections discuss the results of the catenary analysis. 39 Maximum & Minimum Forces & Moments Maximum Hoop Force vs Y/L (Catenary) -3 -2 -1 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Y/L M ax im u m ho o p fo rc e [kN /m ] Fixed Base Ring Beam Base Pinned Base Figure 2.15 ? Graph of Maximum Hoop Force vs. Y/L - Catenary As seen in figure 2.15, the difference between the maximum hoop forces with a pinned base and a fixed base are very small, especially with higher values of Y/L. When a ring beam is used the hoop forces and moments increase considerably. Maximum Meridian Moment vs Y/L (Catenary) -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Y/L M a x im u m m e rid ia n m o m e n t [kN m /m ] Fixed Base Ring Beam Base Pinned Base Figure 2.16 ? Graph of Maximum Meridian Moments vs. Y/L - Catenary 1kN/m = 0.06854 kips/ft 1kNm/m = 0.2248 ft k/ft 40 The results show that as the height of the catenary decreases the base moments and the hoop tension forces increase. The Y = 4.92m (16.1 ft) catenary shape yields the best results for this criterion. This would be a reasonable shape from ground level (option (A)). Reasonably sized shapes for structure type (C) are the Y = 1.88m (6.2 ft) or y = 1.57m (5.1 ft) catenaries. The Tension Region The tensile stresses in the best catenary shape (Y/L = 0.77) are shown in figures 2.17 and 2.18. Hoop Stresses - (Y = 4.92m) -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0 0.5 1 1.5 2 2.5 3 3.5 x - distance from the center of the shape [m] N( ?)/ A + /- M (?) /Z - [M Pa ] Fixed Base Ring Beam Base Pinned Base COMPRESSION TENSION TOP OF DOME Figure 2.17 ? Graph of Hoop Stresses ? Catenary Y = 4.92m (16.1 ft) Meridian Stresses - (Catenary Y = 4.92m) -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0 0.5 1 1.5 2 2.5 3 3.5 x - distance from the center of the shape [m] N( ? )/A + /- M (? )/Z - [M Pa ] Fixed Base Ring Beam Base Pinned Base COMPRESSION TOP OF DOME Figure 2.18 ? Graph of Meridian Stresses ? Catenary Y = 4.92m (16.1 ft) 1m = 3.28 ft 1MPa = 145 psi 1m = 3.28 ft 1MPa = 145 psi 41 The graphs shown on the previous page illustrate the effectiveness of the catenary shape under loading. In the meridian direction the entire structure is in compression. In the hoop direction, a very small portion of the structure is in tension when a ring beam is used. Kicking Out Forces These values were obtained from the fixed base analysis of the structure (refer to figure 2.12 for definition of the variables). Height [m] RF1 (kN/m) RF2 (kN/m) Horizontal Reaction Vertical Reaction (Y = 4.92 m) (16.1 ft) 4.07 (0.28 kip/ft) 15.80 (1.08 kip/ft) (Y = 3.16 m) (10.4 ft) 4.63 (0.32 kip/ft) 11.16 (0.76 kip/ft) (Y = 1.88 m) (6.2 ft) 5.95 (0.41 kip/ft) 8.37 (0.57 kip/ft) (Y = 1.57 m ) (5.1 ft) 6.63 (0.45 kip/ft) 7.83 (0.54 kip/ft) (Y = 1.35 m ) (4.4 ft) 7.32 (0.50 kip/ft) 7.49 (0.51 kip/ft) (Y = 1.19 m ) (3.9 ft) 8.00 (0.55 kip/ft) 7.26 (0.5 kip/ft) Table 2.1? Fixed Base Reaction Forces ? Catenary It can be seen from table 2.1, the kicking out force (RF1) increases as the catenary?s height decreases. As the height of the catenary decreases the shape becomes shallower and the direction of N?(?) tends closer to the horizontal, which increases the horizontal thrusts. Therefore, the shallower shapes require a ring beam with a greater amount of reinforcing to resist the kicking out forces at the dome base. The Best Shapes From the results, it can be seen that the higher the catenary the smaller the kicking out forces, moments and tensile stresses in the structure. This poses a problem as a reasonable height of structure should be chosen for constructability reasons. This is why a maximum value of Y/L = 0.4 is chosen for Type C domes. 42 From the previous analysis, the following optimum Y/L ratios are suggested: Structure Type (A) ? Y/L > 0.5 Structure Type (C) ? Y/L < 0.4 2.6.2 Sections through the Hemisphere The equation for a circle can be used to define the shape of a hemisphere. Once the shape is defined, sections can be taken through hemispheres with different radii. The equation for a circle is: (2.21) where, r is the radius of the circle and x and y are the Cartesian coordinates. The width of the shape was set at 6.4m (21 ft). Five different shaped hemispheres were investigated: r = 3.2m (10.5 ft), r = 3.5m (11.5 ft), r = 4m (13.1 ft), r = 4.5m (14.8 ft) & r = 5m (16.4ft). Maximum & Minimum Forces & Moments Maximum Hoop Force vs Y/L (Hemisphere) -4 -2 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 Y/L M ax im u m ho o p fo rc e [kN /m ] Fixed Base Ring Beam Base Pinned Base Figure 2.19 ? Graph of Maximum Hoop Force vs. Y/L ? Hemisphere 222 ryx =+ 1kN/m = 0.06854 kips/ft 43 Maximum Meridian Moment vs Y/L (Hemisphere) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 Y/L M ax im u m m er id ia n m o m en t [k N m /m ] Fixed Base Ring Beam Base Pinned Base Figure 2.20 ? Graph of Maximum Meridian Moments vs. Y/L ? Hemisphere For pinned and fixed base analyses the results show that an optimum range exists where the maximum hoop forces and meridian moments are minimal. This range can be seen to be between Y/L = 0.24 and 0.32. The inclusion of a ring beam increases both the hoop forces and meridian moments. When a full hemisphere is placed on a ring beam, the hemisphere behaves as if it is pinned at its base (this is in accordance with classical dome/ring theory). The Tension Region The tensile stresses in the best sectioned shape (Y/L = 0.32) are shown below. Hoop Stresses - (Sectioned Hemisphere r=3.5 ; Y=2.08m) -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0 0.5 1 1.5 2 2.5 3 3.5 x - distance from the center of the shape [m] N( ?)/ A + /- M (?) /Z - [M Pa ] Fixed Base Ring Beam Base Pinned Base TENSION COMPRESSION TOP OF DOME Figure 2.21 ? Graph of Hoop Stresses ? Sectioned Hemisphere r = 3.5 1m = 3.28 ft 1MPa = 145 psi 1kNm/m = 0.2248 ft k/ft 44 Meridian Stresses - (Sectioned Hemisphere r=3.5 ; Y=2.08m) -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0 0.5 1 1.5 2 2.5 3 3.5 x - distance from the center of the shape [m] N( ? )/A + /- M (? )/Z - [M Pa ] Fixed Base Ring Beam Base Pinned Base TOP OF DOME TENSION COMPRESSION Figure 2.22 ? Graph of Meridian Stresses ? Sectioned Hemisphere r = 3.5 The hemispherical shape can be sectioned in such a way as to minimize the tension that exists in the structure. The above shape has a Y/L ratio of 0.32 which is in the range mentioned in the previous section. With a fixed base there is no hoop tension in the structure. When the base is pinned there is a small amount of tension in the hoop direction. When a ring beam is placed at the base, tension occurs in the lower part of the structure. Kicking Out Forces r Value RF1 (kN/m) RF2 (kN/m) N'(?) (kN/m) Horizontal Reaction Vertical Reaction 3.2 2.11 (0.14 kip/ft) 12.86 (0.88 kip/ft) 12.86 (0.88 kip/ft) 3.5 4.58 (0.31 kip/ft) 8.89 (0.61 kip/ft) 9.73 (0.67 kip/ft) 4 6.06 (0.42 kip/ft) 8.00 (0.55 kip/ft) 10.00 (0.69 kip/ft) 4.5 7.34 (0.50 kip/ft) 7.73 (0.53 kip/ft) 10.85 (0.74 kip/ft) 5 8.68 (0.59 kip/ft) 7.63 (0.52 kip/ft) 11.91 (0.82 kip/ft) Table 2.2? Fixed Base Reaction Forces ? Sectioned Hemisphere Refer to Figure 2.12 for the definition of the above variables. Table 2.2 shows that the flatter the dome the greater the kicking out forces (RF1). This is because the component of the meridian membrane force in the 1m = 3.28 ft 1MPa = 145 psi 45 horizontal direction increases as the shape becomes shallower. The RF1 value for the full hemisphere is equal to the shear force at the base. This shows that even if the dome meets the wall vertically, there will always be a small amount of kicking out at the intersection of the wall and the dome. A ring beam may not be needed in the case of a full hemisphere as the wall will most probably be able to withstand the small outward shear force. This can be seen in section 2.7 for structure type (B). The Best Shapes The sectioned hemisphere has an optimum range where the membrane forces, moments and shears in the shape are at their least. However, this range yields shallow shapes (in this project as L = 6.4 m (21 ft)) that can only be used if a cylinder wall is placed below the dome. Therefore, a full hemisphere is recommended from ground level from a useable space point of view. Structure Type (A) ? Y/L = 0.5 (Hemisphere) Structure Type (C) ? 0.24< Y/L < 0.32 2.6.3 Sections through the Parabola Six different parabolas were investigated. The variable that was changed in order to obtain the different shapes was the height of the parabola y. The equation of the parabola is: (2.22) Where x and y are the Cartesian coordinates of the parabola. The six parabolas investigated were limited to a base width of 6.4m and their heights were y = 1.2 (3.94 ft), y = 1.6m (5.25 ft), y = 2m (6.56 ft), y =3.2m (10.50 ft), y = 6m (19.68 ft) and y =8m (26.24 ft). Ayx 42 = 46 Maximum & Minimum Forces & Moments Maximum Hoop Force vs Y/L (Parabola) -4 -3 -2 -1 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Y/L M ax im u m ho o p fo rc e [k N /m ] Fixed Base Ring Beam Base Pinned Base Figure 2.23 ? Graph of Maximum Hoop Force vs. Y/L ? Parabola Maximum Meridian Moment vs Y/L (Parabola) -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Y/L M ax im u m m er id ia n m o m en t [ kN m /m ] Fixed Base Ring Beam Base Pinned Base Figure 2.24 ? Graph of Maximum Meridian Moment vs. Y/L ? Parabola The parabola shape makes a very efficient shell structure. The hoop forces when a pinned or fixed base is used are always compressive. The inclusion of a ring beam increases the hoop forces and meridian moments as in the previous shapes. An efficient range where hoop forces and meridian moments are at a minimum exists where 0.8 < Y/L < 1.2. This range corresponds to a height of 1kN/m = 0.06854 kips/ft 1kNm/m = 0.2248 ft k/ft 47 between 5 (16.4 ft) and 8 meters (26.2 ft). It is interesting to note that the Musgum tribe of Central Africa built parabolic mud huts (figure 1.4) within the same height range. The Tension Region The tensile stresses in the best parabolic shape (Y/L = 1.2) are presented below. Hoop Stresses (Parabola y=8m) -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0 0.5 1 1.5 2 2.5 3 3.5 x - distance from the center of the shape [m] N( ?)/ A + /- M (?) /Z - [M Pa ] Fixed Base Ring Beam Base Pinned Base TENSION COMPRESSION TOP OF DOME Figure 2.25 ? Graph of Hoop Stresses ? Parabola (y = 8m; 26.2 ft) Meridian Stresses (Parabola y=8m) -0.2 -0.18 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0 0.5 1 1.5 2 2.5 3 3.5 x - distance from the center of the shape [m] N( ? )/A + /- M (? )/Z - [M Pa ] Fixed Base Ring Beam Base Pinned Base TOP OF DOME COMPRESSION Figure 2.26 ? Graph of Meridian Stresses ? Parabola (y = 8m; 26.2 ft) 1m = 3.28 ft 1MPa = 145 psi 1m = 3.28 ft 1MPa = 145 psi 48 Once again, it can be seen that the inclusion of a ring beam produces a small degree of tension in the hoop direction of the structure. Otherwise, the entire structure is in compression. This was beneficial to the Musgums as their building material was compacted mud, which has almost no tensile strength. Kicking Out Forces y Value RF1 (kN/m) RF2 (kN/m) Horizontal Reaction Vertical Reaction 1.2 8.04 (0.55 kip/ft) 7.26 (0.50 kip/ft) 1.6 6.73 (0.46 kip/ft) 7.83 (0.54 kip/ft)) 2 6.00 (0.41 kip/ft) 8.51 (0.58 kip/ft) 3.2 5.10 (0.35 kip/ft) 10.90 (0.75 kip/ft) 6 4.81 (0.33 kip/ft)) 17.50 (1.20 kip/ft) 8 4.90 (0.34 kip/ft) 22.54 (1.54 kip/ft) Table 2.3 ? Parabola ? Fixed Base Reaction Forces Refer to Figure 2.12 for the definition of the above variables. The values of the kicking out forces for parabolas are very similar to the values presented for the catenary shape. This is expected, as the two shapes are very similar. The Best Shapes An optimum range for parabolic shaped shell structures is 0.8 < Y/L < 1.2. This range produces very high structures which would only be viable if built from ground level (structure type A). For structure type C, the parabola is not recommended. 49 2.6.4 The Ellipse The advantage of an elliptical shape is that the roof will meet the wall vertically, and therefore the meridian membrane force is transmitted vertically into the wall. The only contribution to the kicking out force will be the shear in the section and a ring beam may not be required for structure types (A) and (B). However, an elliptical roof is very flat at the top and causes difficulties in the construction of this shape. Three different ellipses were investigated. The variable that was changed in order to obtain the different shapes was the height of the ellipse, B. The equation of the ellipse is: 1)()( 2 2 2 2 =+ B y H x (2.23) Where x and y are the Cartesian coordinates of the ellipse. B is the height along the minor axis and H the height along the major axis. The four ellipses investigated were limited to a base width of 6.4m (21 ft) and their heights were B = 1.6m (5.25 ft), B = 1.8m (5.90 ft), B =2m (6.56 ft) and B=3.2m (10.50 ft) (hemisphere). 50 Maximum & Minimum Forces & Moments Maximum Hoop Force vs Y/L (Ellipse) 0 2 4 6 8 10 12 14 16 0 0.1 0.2 0.3 0.4 0.5 0.6 Y/L M ax im u m ho o p fo rc e [kN /m ] Fixed Base Ring Beam Base Pinned Base Figure 2.27 ? Graph of Maximum Hoop Force vs. Y/L ? Ellipse Maximum Meridian Moment vs Y/L (Ellipse) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 Y/L M a xi m u m m e rid ia n m o m en t [ kN m /m ] Fixed Base Ring Beam Base Pinned Base Figure 2.28 ? Maximum Meridian Moment vs. Y/L ? Ellipse Figure 2.27 and figure 2.28 show that as the height of the ellipse increases the forces and moments in the ellipse decrease. The B = 3.2m (10.5 ft) ellipse (hemisphere) is the best alternative for this criterion. A considerable amount of hoop tension exists at the base of an ellipse. This is undesirable for shell structures. 1kN/m = 0.06854 kips/ft 1kNm/m = 0.2248 ft k/ft 51 Tension Region (Stresses) The tensile stresses in the best elliptical shape (Y/L = 0.5) are presented below. Hoop Stresses (Ellipse r=3.2 ; y=3.2m) -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0 0.5 1 1.5 2 2.5 3 3.5 x - distance from the center of the shape [m] N( ?)/ A + /- M (?) /Z - [M P a] Fixed Base Ring Beam Base Pinned Base TENSION COMPRESSION TOP OF DOME Figure 2.29 ? Graph of Hoop Stresses ? Ellipse (B = 3.2m; Y = 3.2m) Meridian Stresses (Ellipse r=3.2 ; y=3.2m) -0.25 -0.2 -0.15 -0.1 -0.05 0 0 0.5 1 1.5 2 2.5 3 3.5 x - distance from the center of the shape [m] N( ? )/A + /- M (? )/Z - [M Pa ] Fixed Base Ring Beam Base Pinned Base TOP OF DOME COMPRESSION Figure 2.30 ? Graph of Meridian Stresses ? Ellipse (B = 3.2m; Y = 3.2m) 1m = 3.28 ft 1MPa = 145 psi 1m = 3.28 ft 1MPa = 145 pi 52 Kicking Out Forces y Value RF1 (kN/m) RF2 (kN/m) Horizontal Reaction Vertical Reaction 1.2 6.28 (0.43 kip/ft) 8.01 (0.55 kip/ft) 1.6 4.37 (0.30 kip/ft) 8.86 (0.61 kip/ft) 2 3.32 (0.23 kip/ft) 9.78 (0.67 kip/ft) 3.2 2.11 (0.14 kip/ft)) 12.86 (0.88 kip/ft) Table 2.4? Fixed Base Reaction Forces ? Ellipse Refer to Figure 2.12 for the definition of the above variables. The kicking out force for the ellipse structure is the same magnitude as the shear force at the base of the ellipse. The above results show that as the ellipse approaches the shape of the hemisphere the kicking out force reduces. The Best Shapes The ellipse is a very inefficient shape and it is not recommended. The most efficient ellipse is the hemisphere, which is a suitable structure for structure type A. 2.7 Shape Analysis Results (Structure Type B) The shapes investigated in this section were placed on top of a 1m high cylindrical wall in order to improve useable space within the structure. No ring beam was included in this analysis as the wall was assumed to resist the lateral thrust of the dome. The overall height of the structure was limited to approximately four meters and the diameter to 6.4m (21 ft). The hoop force and meridian moment diagrams are presented on the next page. 53 Hoop Forces (Shell with 1m cylinder wall ) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 -10 -5 0 5 10 15 20 Force [kN/m] He ig ht [m ] Catenary Hemisphere Parabola WALL HEIGHT 1m TOP OF DOME Figure 2.31 ? Graph of N(?) ? Shells with 1m Cylinder Walls ? Structure Type B Meridian Moments (Shell with 1m cylinder wall) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Mome nt [kNm/m] He ig ht [m ] Catenary Hemisphere Parabola WALL HEIGHT 1m TOP OF DOME Figure 2.32 ? Graph of M(?) ? Shells with 1m Cylinder Walls ? Structure Type B 1m = 3.28 ft 1kN/m = 0.06854 kips/ft 1m = 3.28 ft 1kNm/m = 0.2248 ft k/ft 54 From figure 2.31 and 2.32 it can be seen that the hemisphere placed on top of a cylinder wall is the best option for structure type B. The kicking out force is the least for the hemisphere option and it has the most useable space within the structure. It can be concluded that the best shape that can be placed on a cylinder wall is one that meets the wall vertically (or near vertically). This is because there is no component of the meridian membrane force in the horizontal direction at the dome/cylinder wall interface, which would kick out the cylinder wall causing tension in the structure. This is the same argument as that presented in section 2.6.2 (Kicking out Forces). However, a shear force and a moment still exist at this interface and this produces some outward movement of the wall. Therefore the moments and hoop forces obtained from this analysis are greater than the results obtained in an analysis of a hemisphere pinned at its base. The problems with this type of structure are: ? The hoop forces and meridian moments are much larger than a shell from ground level. ? The maximum forces and moments occur in regions where door and window openings will be placed in the structure. These openings will increase the stresses in the structure further. 2.8 Summary of the Results 2.8.1 Tabulated Summary This section summarizes the results of the best performing structures. It also compares the useable space within the selected optimum structures. The parameters used in measuring the useable space within the structure are shown in figure 2.33, overleaf. 55 USEABLE SPACE - Definition of Variables 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Distance from Centre of Section (m) Surface of Shape Diameter of Useable Floor Area Door Height (2.1m) Figure 2.33 ? Useable Space Parameters Structure {A} Structure {B} Structure {C} (i) Dome from ground level (pinned base) Catenary Hemisphere Parabola Shape Evaluation Criteria Units a = 1.5 r = 3.2 y = 8 (ii) Hemisphere on 1m wall (pinned base) (iii) Sectioned Hemisphere (r = 3.5) on a ring beam Y/L .1. Optimum Y/L Range - > 0.5 0.5 0.8 0.5 ? Structure Type B ? Hemisphere ? Structure Type C ? Sectioned Hemisphere ? 0.24 < Y/L < 0.32 Important Design Criteria for Masonry Domes In the analysis section of this report it was found that the most critical areas in domes are around the window openings. In these regions, high tension forces and moments occur in both meridian and hoop directions. In masonry structures tension is critical and should be minimized or avoided if possible. From the analysis it was found that the inclusion of arches in window and door openings reduces the tension in these regions. The arches stiffen the structure. However, under temperature loading, it was found that some form of reinforcing (in both meridian and hoop directions) was required in these regions. In the prototype structure that was built another critical region was recognized. This was the bottom third of the hemisphere (measured from the intersection of the dome with the cylinder wall). Significant hoop tension exists in this region and reinforcing (wire mesh) is recommended. The most critical load cases were 1.5DL (for compression resistance), 0.9DL + 1.3 WL (suction side ? for hoop tension in the dome) and 0.9DL + 1.2 TL (for tension around the openings). Wind Loading The hoop tension in the dome (centre section) was critical for this load case. However, the 1.5DL load case produced very similar results to this load case and it can be assumed that wind load has a small effect on the structure. This 148 structural efficiency under wind loading is one of the major benefits of dome structures. Temperature Loading The fixity of the base of a dome is important when temperature load is being considered. If the base is fixed and there is a temperature differential between the dome and the foundation, large hoop forces and moments (hoop & meridian) will occur towards the base of the dome. In this investigation, a DPC was provided at this interface providing a slip plane. This plane is important if reinforcing in this region is to be avoided. It is recommended that when a structural analysis is done on a dome structure particular attention should be given to the openings within the structure as well as temperature loading on the structure. The openings should be rounded to reduce stress concentrations and arches should be built within the openings to stiffen the structure. Materials Investigated for Dome Construction The materials investigated for use in the dome were cement stabilized earth blocks, wire mesh, wire wrapping (stitching around openings) and fibre reinforced plaster. Materials Resisting Compression The cement stabilized earth blocks proved to be a good alternative to standard clay or concrete bricks. Their thermal properties are more favorable than standard bricks and they can be made on site in the most rural areas of Southern Africa. 149 Materials Resisting Tension (Cracking) The wire mesh and fibre plaster were investigated for their potential to resist tension and stop any cracking of the structure. The fibre plaster was not used because the discontinuous nature of the fibres, which allows cracks to spread once they have started. The wire mesh, on the other hand, intersects the crack and prevents it from spreading.. Wire wrapping around highly stressed regions (openings) was used in order to resist tension in these regions. The wire wrapping was placed perpendicular to potential cracks observed in previously built dome structures, which matched the high stress regions shown by the finite element analysis. This type of reinforcing allows the tension regions to be targeted and it minimizes the wastage of expensive tension resisting materials. It is recommended that wire mesh be used on the inside and outside faces of the dome structure in order to resist the small tensile forces within the structure as well as to prevent plaster cracking (Williams-Ellis, 1947). The effectiveness of the wire wrapping around the openings in the structure is still being observed but this method of reinforcing could possibly replace the inefficient use of Brickforce in the structure. It is important to note that no cracking has occurred on the prototype structure to date. Methods of Construction Two viable methods of construction of the dome were identified in this report. The first method was the Nubian (tracing arm) method of construction and the second was the construction of the dome using an inflatable formwork. The inflatable formwork and its associated equipment may be difficult to use in rural areas for small projects and for these types of projects it is advisable that the Nubian technique be used. This technique may be more time consuming than the inflatable formwork method but it has been proven successful in India 150 at the Auroville Institute and could be used by rural communities to build their own homes. The inflatable formwork method of construction is an efficient method which saves time and can be effectively used on a mass scale as the cost of the equipment is spread over many houses. A greater amount of skill and supervision is needed for this method of construction. The dome structure has proved to be a cost effective structure in Mozambique and the United States. The dome built in this project was not as competitive as these structures with regard to cost (high labour costs). However, in the cost analysis this structure was the only dome structure to make provision for services and the quality of the structure was very good. Further studies into construction techniques are recommended. Dome and vault roof structures built on square floor plans could also be investigated as these structures would provide a greater amount of useable space than circular floor plans. 151 Appendix A (Selected AbaqusTM Output) The results of the analysis done on the prototype 28m2 (301 ft2) dome are presented in tabular format below. The results are presented along the centre section as seen in figure 4.7. The load cases that were critical for this cross- section were the 1.5DL and 0.9DL+1.3WL load cases. For localized results around the window and door openings see sections 4.4.2 and 5.2.1. 1.5DL X Y Hoop Force [kN/m] Meridian Moment [kNm/m] Meridian Force [kN/m] Hoop Moment [kNm/m 0.205 3.762 -5.915 -0.009 -5.759 -0.011 0.306 3.753 -6.070 -0.009 -5.478 -0.012 0.406 3.740 -6.000 -0.008 -5.440 -0.012 0.506 3.723 -5.930 -0.008 -5.417 -0.012 0.605 3.703 -5.862 -0.008 -5.388 -0.012 0.703 3.679 -5.795 -0.008 -5.351 -0.012 0.800 3.652 -5.724 -0.008 -5.306 -0.012 0.897 3.621 -5.649 -0.008 -5.256 -0.012 0.992 3.586 -5.569 -0.007 -5.200 -0.012 1.085 3.548 -5.480 -0.007 -5.140 -0.011 1.178 3.507 -5.383 -0.007 -5.078 -0.011 1.268 3.463 -5.276 -0.007 -5.014 -0.010 1.357 3.415 -5.157 -0.008 -4.950 -0.009 1.445 3.363 -5.025 -0.008 -4.888 -0.009 1.530 3.309 -4.876 -0.009 -4.828 -0.008 1.613 3.252 -4.709 -0.010 -4.774 -0.007 1.694 3.191 -4.519 -0.011 -4.726 -0.006 1.773 3.128 -4.304 -0.012 -4.687 -0.004 1.850 3.062 -4.060 -0.014 -4.659 -0.004 1.924 2.993 -3.783 -0.016 -4.643 -0.003 1.995 2.922 -3.469 -0.017 -4.642 -0.002 2.064 2.847 -3.116 -0.018 -4.656 -0.002 2.130 2.771 -2.719 -0.019 -4.689 -0.002 2.193 2.692 -2.276 -0.020 -4.741 -0.002 2.253 2.611 -1.784 -0.019 -4.817 -0.003 2.311 2.527 -1.242 -0.018 -4.917 -0.004 2.365 2.442 -0.649 -0.015 -5.047 -0.005 2.416 2.355 -0.006 -0.011 -5.211 -0.006 2.464 2.266 0.683 -0.006 -5.411 -0.008 2.508 2.175 1.411 0.001 -5.654 -0.009 2.550 2.083 2.165 0.010 -5.942 -0.009 2.587 1.989 2.928 0.020 -6.278 -0.009 152 X Y Hoop Force [kN/m] Meridian Moment [kNm/m] Meridian Force [kN/m] Hoop Moment [kNm/m 2.622 1.894 3.673 0.032 -6.664 -0.007 2.653 1.798 4.367 0.045 -7.098 -0.006 2.680 1.700 4.968 0.058 -7.579 -0.002 2.704 1.602 5.431 0.071 -8.098 0.001 2.724 1.503 5.708 0.081 -8.650 0.005 2.741 1.403 5.756 0.088 -9.226 0.008 2.753 1.303 5.544 0.086 -9.818 0.011 2.763 1.202 5.070 0.072 -10.419 0.011 2.768 1.101 4.363 0.043 -11.021 0.008 2.770 1.000 9.878 -0.012 -11.619 0.001 2.77 0.900 7.839 -0.067 -12.572 -0.006 2.77 0.800 5.962 -0.098 -13.564 -0.007 2.77 0.700 4.263 -0.107 -14.593 -0.005 2.77 0.600 2.736 -0.100 -15.652 -0.001 2.77 0.500 1.358 -0.082 -16.731 0.003 2.77 0.400 0.098 -0.058 -17.825 0.006 2.77 0.300 -1.080 -0.032 -18.926 0.008 2.77 0.200 -2.211 -0.010 -20.031 0.009 2.77 0.100 -3.326 0.004 -21.136 0.008 2.77 0.000 -4.447 0.002 -22.239 0.008 Table A1 ? 1.5DL ? Centre Section Results 0.9DL+1.3WL (Suction Side) X Y Hoop Force [kN/m] Meridian Moment [kNm/m] Meridian Force [kN/m] Hoop Moment [kNm/m 0.205 3.762 -3.559 -0.005 -3.364 -0.008 0.317 3.752 -3.512 -0.005 -2.986 -0.008 0.429 3.737 -3.421 -0.005 -2.917 -0.008 0.540 3.717 -3.309 -0.004 -2.859 -0.008 0.647 3.693 -3.191 -0.004 -2.776 -0.007 0.752 3.666 -3.082 -0.004 -2.694 -0.007 0.856 3.634 -2.972 -0.004 -2.608 -0.008 0.959 3.599 -2.862 -0.005 -2.518 -0.007 1.060 3.559 -2.744 -0.004 -2.424 -0.007 1.160 3.516 -2.623 -0.004 -2.330 -0.006 1.258 3.468 -2.497 -0.004 -2.237 -0.006 1.353 3.417 -2.364 -0.005 -2.146 -0.005 1.447 3.362 -2.223 -0.006 -2.056 -0.005 1.539 3.303 -2.067 -0.006 -1.968 -0.004 1.628 3.241 -1.900 -0.006 -1.887 -0.003 1.715 3.175 -1.717 -0.008 -1.812 -0.002 1.799 3.106 -1.516 -0.009 -1.746 -0.001 1.880 3.034 -1.294 -0.010 -1.690 -0.001 153 X Y Hoop Force [kN/m] Meridian Moment [kNm/m] Meridian Force [kN/m] Hoop Moment [kNm/m 1.959 2.959 -1.044 -0.011 -1.644 0.000 2.034 2.880 -0.769 -0.012 -1.612 0.000 2.106 2.799 -0.463 -0.012 -1.594 0.000 2.175 2.715 -0.126 -0.013 -1.593 -0.001 2.241 2.628 0.245 -0.012 -1.612 -0.002 2.303 2.539 0.656 -0.011 -1.650 -0.003 2.362 2.447 1.103 -0.008 -1.717 -0.004 2.417 2.353 1.588 -0.005 -1.811 -0.006 2.468 2.258 2.108 0.000 -1.941 -0.007 2.516 2.160 2.661 0.005 -2.112 -0.009 2.559 2.060 3.233 0.012 -2.321 -0.009 2.599 1.959 3.814 0.020 -2.584 -0.008 2.634 1.856 4.375 0.029 -2.890 -0.007 2.666 1.752 4.893 0.039 -3.245 -0.005 2.693 1.647 5.328 0.048 -3.642 -0.002 2.717 1.540 5.618 0.057 -4.075 0.002 2.736 1.433 5.775 0.063 -4.551 0.006 2.751 1.326 5.720 0.065 -5.049 0.010 2.761 1.217 5.446 0.059 -5.569 0.012 2.768 1.109 4.974 0.043 -6.103 0.010 2.770 1.000 10.540 0.008 -6.643 0.040 2.770 0.900 8.800 -0.025 -7.376 0.032 2.770 0.800 7.196 -0.040 -8.159 0.033 2.770 0.700 5.731 -0.038 -8.984 0.034 2.770 0.600 4.382 -0.026 -9.842 0.036 2.770 0.500 3.120 -0.007 -10.724 0.036 2.770 0.400 1.910 0.012 -11.622 0.035 2.770 0.300 0.716 0.029 -12.530 0.032 2.770 0.200 -0.494 0.038 -13.442 0.026 2.770 0.100 -1.744 0.032 -14.353 0.019 2.770 0.000 -3.047 0.003 -15.259 0.014 Table A2 ? 0.9DL+1.3WL (Suction side) ? Centre Section Results 154 Appendix B (Alternative Structure) The alternative structure, presented below (figure B1), is a sectioned hemisphere with a Y/L ratio of 0.3. Structurally and architecturally this structure was more efficient than the dome constructed. However, the projected cost of this structure was greater than the one built. Figure B1 ? Alternative Structure ? Structure Type C The forces and moments in the dome and cylinder wall are presented in the following figures. 1m = 3.28 ft 155 The Dome Results Meridian Forces (ULS) - Dome -12 -10 -8 -6 -4 -2 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Distance from the centre of the dome [m] Fo rc e [kN /m ] 1.5DL 1.2DL+1.6LL (Point Load @ Skylight) 1.2DL+1.6LL (Point Load 1/3rd Up Dome) 0.9DL+1.3WL (Suction Side) 0.9DL+1.3WL (Pressure Side) 1.2TL+0.9DL (+24?) TOP OF DOME Figure B2 ? Meridian Forces (Dome) ? Structure Type C Hoop Forces (ULS) - Dome -20 -15 -10 -5 0 5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Distance from the centre of the dome [m] Fo rc e [kN /m ] 1.5DL 1.2DL+1.6LL (Point Load @ Skylight) 1.2DL+1.6LL (Point Load 1/3rd Up Dome) 0.9DL+1.3WL (Suction Side) 0.9DL+1.3WL (Pressure Side) 1.2TL+0.9DL (+24?) TOP OF DOME Figure B3 ? Hoop Forces (Dome) - Structure Type C 1kN/m = 0.06854 kips/ft 1kN/m = 0.06854 kips/ft 156 Meridian Moments (ULS) - Dome -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Distance from the centre of the dome [m] M o m e n t [k N m /m ] 1.5DL 1.2DL+1.6LL (Point Load @ Skylight) 1.2DL+1.6LL (Point Load 1/3rd Up Dome) 0.9DL+1.3WL (Suction Side) 0.9DL+1.3WL (Pressure Side) 1.2TL+0.9DL (+24?) TOP OF DOME Figure B4 ? Meridian Moments (Dome) ? Structure Type C The Cylinder Wall Results Meridian Forces (ULS) - Cylinder Wall 0 1 1 2 2 3 -35 -30 -25 -20 -15 -10 -5 0 Forces [kN/m] He ig ht [m ] 1.5DL (Pinned/Sliding) 1.2DL+1.2TL (+24 degrees) (Fixed) 0.9DL+1.3WL (Pinned) Suction Side 1.2DL+1.2TL (+24 degrees) (Sliding) 0.9DL+1.3WL (Pinned) Pressure Side Figure B5 ? Meridian Forces (Cylinder) ? Structure Type C 1kNm/m = 0.2248 ft k/ft 1kN/m = 0.06854 kips/ft 157 Hoop Forces (ULS) - Cylinder Wall 0.0 0.5 1.0 1.5 2.0 2.5 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 Force [kN/m] He ig ht [m ] 1.5DL (Pinned/Sliding) 0.9DL+1.3WL (Pinned) Suction Side 1.2DL+1.2TL (+24 degrees) (Sliding) 0.9DL+1.3WL (Pinned) Pressure Side TOP OF WALL Figure B6 ? Hoop Forces (Cylinder) ? Structure Type C Meridian Moments (ULS) - Cylinder Wall 0.0 0.5 1.0 1.5 2.0 2.5 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 Moment [kNm/m] He ig ht [m ] 1.5DL (Pinned/Sliding) 0.9DL+1.3WL (Pinned) Suction Side 1.2DL+1.2TL (+24 degrees) (Sliding) 0.9DL+1.3WL (Pinned) Pressure Side TOP OF WALL Figure B7 ? Meridian Moments (Cylinder) ? Structure Type C These forces and moments were well within the allowable stress zone of an uncracked masonry analysis. The absence of openings in the dome (except for the skylight) improved the efficiency of the dome. The ring beam at the base of the dome roof acts like a lintel for the window openings in the cylinder wall. A similar design to this was used at the Thholego Eco-Village. 1kNm/m = 0.2248 ft k/ft 1kN/m = 0.06854 kips/ft 158 References Addis, B. et al (2001) Fulton?s Concrete Technology: Fibre Reinforced Concrete, Cement and Concrete Institute, Midrand, South Africa. 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