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Form-Finding of Arch Structures A Thesis Submitted for the Degree of PhD at the University of Warwick Permanent WRAP URL: http://wrap.warwick.ac.uk/87332 Copyright and reuse: This thesis is made available online and is protected by original copyright. Please scroll down to view the document itself. Please refer to the repository record for this item for information to help you to cite it. Our policy information is available from the repository home page. For more information, please contact the WRAP Team at: [email protected] warwick.ac.uk/lib-publications Form-finding of Arch Structures By Asal Pournaghshband A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Engineering University of Warwick, School of Engineering May 2016 Table of Contents List of Figures .......................................................................................................... vi List of Tables ........................................................................................................... x Acknowledgements ................................................................................................ xiv Declaration ............................................................................................................. xv Abstract ................................................................................................................. xvi Symbols ................................................................................................................ xvii Abbreviations .......................................................................................................... xx Chapter 1 : Introduction ........................................................................................... 1 1.1. Framework and Motivation......................................................................... 1 1.2. Research aims and objectives ................................................................... 4 1.3. Structure of the thesis ................................................................................ 6 Chapter 2 : Literature review .................................................................................... 8 2.1. Introduction ................................................................................................... 8 2.2. History of arches ........................................................................................... 9 2.3. Study of the behaviour of arches ................................................................. 10 2.4. Study of the analysis methods of arches ..................................................... 14 2.5. Optimization and form-finding survey .......................................................... 35 2.6. Concluding remarks .................................................................................... 54 Chapter 3 : Analysis of two-pin arches ................................................................... 57 3.1. Introduction ................................................................................................. 57 3.2. Theory of common arch shapes .................................................................. 57 i 3.2.1. Derivation of arch equations ................................................................. 58 3.2.1.1. Circular form equation ........................................................................ 58 3.2.1.2. Parabolic form equation ..................................................................... 59 3.2.1.3. Catenary form equation...................................................................... 60 3.3. Project description ....................................................................................... 61 3.3.1. Arch specifications ................................................................................ 61 3.3.2 Load cases ............................................................................................ 62 3.3.3. Assessment of arch stability over considered L:h ratios ........................ 65 3.4. Assessment criterion ................................................................................... 68 3.5. Methodology ................................................................................................ 69 3.5.1. Bending moments ................................................................................. 70 3.5.2. Combined stresses ............................................................................... 78 3.5.3. Compressive force (thrust) .................................................................... 80 3.5.4. Displacements and deflections.............................................................. 81 3.5.5. Shear forces ......................................................................................... 84 3.5.6. Horizontal reaction force ....................................................................... 85 3.5.7. Arch mass ............................................................................................. 86 3.6. Arches of steel material ............................................................................... 87 3.7. Exploring the effect of different ratios of UDL:SW ........................................ 88 3.8. Exploring the effect of arch thickness .......................................................... 92 3.9. Exploring the effect of shortening of the arch ............................................... 95 3.9.1. Comparison of horizontal reaction force from GSA with other methods . 95 ii 3.9.2. Threshold range of L:h ratio of arch shallowness .................................. 99 3.10. Concluding Remarks ............................................................................... 100 Chapter 4 : Comparison of different methods of two-pin arch analysis ................. 102 4.1. Introduction ............................................................................................... 102 4.2. Theoretical analysis method ...................................................................... 103 4.3. Approximate analytical methods ................................................................ 105 4.3.1. Masonry design method ...................................................................... 105 4.3.2. Virtual work method presented by Megson ......................................... 107 4.4. Comparison of different methods of analysis ............................................. 110 4.5. Exploring the thickness effect on horizontal reaction force using the Castigliano approach........................................................................................ 123 4.6. Exploring the shortening effect .................................................................. 131 4.6.1. Shortening effect of flat arches............................................................ 132 4.7. Concluding Remarks ................................................................................. 135 Chapter 5 : Analytical form-finding of two-pin arch ............................................... 137 5.1. Introduction ............................................................................................... 137 5.2. Optimality criterion of two-pin arches ......................................................... 139 5.3. Equilibrium of any point on the arch........................................................... 140 5.4. Geometry of the momentless arch ............................................................. 145 5.4.1. Finding the ‘x’ coordinate of the momentless arch .............................. 145 5.4.2. Finding the ‘y’ coordinate of the momentless arch .............................. 147 5.4.3. Finding the shape of momentless arch ................................................ 147 5.4.4. Determination of the initial value of tmax when UDL:SW ≥ 1 ................. 150 iii 5.4.5. Determination of the initial value of tmax when UDL:SW < 1 ................. 150 5.4.6. Determination of horizontal reaction force ........................................... 151 5.5. Comparing the geometry of the momentless arch with known shapes of arches .............................................................................................................. 151 5.6. Case study ................................................................................................ 155 5.6.1. Calculating the weight of the arch rib .................................................. 157 5.6.2. Calculating the weight of the vertical members (piers) ........................ 157 5.6.3. Calculating the weight of the deck (superimposed dead load) ............. 161 5.6.4. Calculating the traffic load ................................................................... 161 5.6.5. Arch model.......................................................................................... 162 5.6.6. Calculating the required reinforcement ................................................ 165 5.6.7. Comparing the deflections of parabolic and momentless arches ......... 168 5.6.8. Comparing the first failure of the cross-sections of the parabolic and momentless arches ....................................................................................... 171 5.7. Concluding Remarks ............................................................................. 174 Chapter 6 : Discussion and Conclusions .............................................................. 176 6.1. Discussion of the general behaviour of the two-pin arch ............................ 176 6.2. Discussion of the comparison of different
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