CONTENTS Foreword XI Introduction 1 The path-way of structures 2 What is a load-bearing structure? 4 The purpose of a structure 4 Structure and architecture 5 Forces and equilibrium, internal forces, strength and stiffness 7 Loads that act on a structure 9 Gravitational forces and Newton's law of gravitation 9 Gravitational force on the surface of the earth 10 Force vectors, point of application and line of action 10 Free body diagram 11 Conditions of equilibrium of two forces 11 Forces that act on the surface of contact between two free bodies: action = reaction. 12 Transmission of a force and internal force 13 The compressive internal force and its quantification 13 Effect of compression on materials: compressive stress 14 The tensile internal force 15 Tensile stress 16 Effect of tension: elongation 16 Effect of compressiomshortening 16 Linear behavior and elastic behavior 17 Stiffness 17 Stiffness of a structure subjected to tension or compression 18 Stiffness of the material. 18 Elastic phase and plastic phase 19 Yield strength and strength. 20 Mechanical behavior of steel 20 Modulus of elasticity E 20 Yield strenghtf 21 Tensile strength ft 21 Strain at failure a. 21 Tension and compression. 22 Fragility and ductility 22 Concrete. 23 Stone 23 Wood 24 Comparison of materials 24 Stiffness and strength 24 Dimensioning. .. 26 Criterion of the serviceability limit state (SLS) 26 Criterion of the ultimate limit state (ULS) 27 Load factors 27 Factored loads and design value of the internal force 27 Resistance factors 27 Design strength 28 Fatigue 29 Equilibrium of more than two forces in a plane and in space. 29 First condition of equilibrium 30 Polygon of forces or force polygon. 30 Second condition of equilibrium 30 Point of application of a force and equilibrium 30 Angle of friction 31 Cremona diagram31 Forces and internal forces 32 Cables 35 Structural diagrams37 The span f: and the rise 37 Supports 38 Direction of the internal force on the subsystem 38 Influence of the load 39 Influence of geometry 39 The f/fratio 40 Influence of the position of the load 40 Load in any direction 41 Cable with two vertical loads41 Resultant cable 42 Cable with two non-vertical load 42 Cable with multiple non-vertical loads 43 Parallel non-symmetrical loads 44 Auxiliary cable44 Center of gravity 45 Funicular polygon 45 Distributed loads 45 Cable subjected to uniformly distributed loads 45 Catenaiy 47 Suspension bridges47 Applications in architecture 48 Supports: pylons, anchors and other elements 49 Sizing of cables 49 Section of the cable in relation to the slendcrness ratio 49 Quantity of material based on the slcnderness ratio 50 Movements caused by variation of the intensity of loads 50 Deflections caused by permanent loads 51 Movements caused by variable loads51 Deflections caused by temperature variation52 Effect of horizontal movements of the supports on the geometry of the cable 52 Variation of the configuration of the loads52 Movements caused by variable loads 53 Limiting displacements caused by variable loads . 53 Increase of the permanent load .54 Solution with stiffening cable: cable beams 54 Solution with load-bearing cable and stabilizing cables 56 The cable with stiffening beam . 57 Cables with flexural stiffness. 58 Systems with combined cables 59 Cable-stayed systems 59 Cable networks, tents and membranes 61 Systems of cables in space 63 Cable networks. . 64 Tents and membranes 65 Pneumatic membranes67 High-pressure pneumatic membranes67 Structures under compression 71 Cases with multiple loads or distributed loads: arches 72 Parabolic arches72 Catenary arches 72 Similarity between cables and arches73 Influence of variable loads 73 Instability of arches 74 Provisions to stabilize arches 74 Addition of stabilizing bars 74 Insertion of a stiffening beam 75 Stiffening the arches by increasing the thickness 75 Line of action of the internal forces 76 Possible lines of action of internal forces inside an arch78 Statically indeterminate and determinate arches 78 Three-hinged arches 79 Optimal form of a three-hinged arch 79 Required thickness of a three-hinged arch subjected only to compression 8O Arches constructed with materials resistant to tension 81 Arches whose form docs not correspond to that of the funicular polygorf^f permanent loads 82 Two-hinged arches, the ideal form 82 Arches with one hinge83 Arches without hinge 83 Semi-circular masonry arches 84 Vaults, domes and shells 87 Arches as roofing elements. 89 Barrel vaults 89 Groin vaults91 Fan vaults 92 Pavilion vaults 94 Domes 95 Effective functioning of domes 95 Domes with a central skylight opening. 96 Steel domes 97 Crossed arches 97 Domes composed of arches and rings98 Shape of domes and the stresses 99 Conical domes 100 Hyperboloids of revolution 101 Carrying of non-symmetrical horizontal or vertical loads 103 Geodesic domes 104 Grid domes104 Shells and arbitrary domes 105 Downward double-curvature shells106 Shells with upward or downward curvature, the hyperbolic paraboloid107 Monkey saddle surfaces108 Composed arbitrary shells 108 Cylindrical shells 109 Gridshcll structures 110 Arch-cables 111 Carrying the horizontal thrust component 113 Arches with ties 114 Fixed and sliding supports114 Design and analysis of arches with ties 115 Cable-strut compositions 116 Arch and cable compositions 117 Arch-cables 118 Stabilization of the arch and carrying of variable loads 118 Arch-cable cantilevers 119 Cable-stayed systems 120 Trusses 123 Solution to the problem of dcformability and stability by the addition of supplementary bars 125 Trusses 125 Analysis of trusses 125 Unstable, statically determinate and statically indeterminate systems 128 Generation of trusses 129 General analysis of trusses129 Upper chord, lower chord and diagonals 132 Influence of the height and the span on the internal forces in trusses132 Complete analysis of a truss133 Bending moments 134 Identification of the bars under largest stress in the chords135 Targeted analysis of bars in the chords of trusses135 Analysis of the diagonals and their functioning 137 Shear force 137 Identification of diagonals with the largest internal force. 138 Identification of diagonals under tension and those under compression. 138 Qualitative analysis of a truss. 139 Possible configurations of the diagonals139 V diagonals.139 N diagonals 140 X diagonals 140 K diagonals142 Forms of trusses145 Form and structural efficiency 146 Influence of form on structural stiffness148 Cantilevers and towers with multiple loads 149 Towers 150 Trusses with cantilevers 151 Gcrber beams152 Trusses for other structural forms 153 Space trusses 155 Composition of trusses to support a roof 157 Lattice trusses 157 Space trusses 158 Vaults and domes composed of trusses 159 Internal forces in the middle zone of beams163 Internal force in zones m tension and m compression 163 Reinforced concrete beams164 Simple bending of a beam 165 Bending and curvature 165 Strength of beams subjected to bending 165 Influence of the dimensions of a rectangular beam on its strength. 166 Influence of the dimensions of a rectangular beam on its stiffness 168 The most efficient sections: wide-flange sections 169 Influence of the dimensions of a wide-flange beam on its bending strength and stiffness 170 Behavior of a wide-flange beam with vertical flanges171 The efficiency of a section 172 Form, section and structural efficiency 172 Simple beams with concentrated and distributed loads 174 Cantilevers 175 Beams with cantilevers 176 Gerber beams178 Continuous beams179 Bi-clampcd beams 180 Zones of greater or lesser internal forces in beams 181 Frames 183 Two-hinged frames 185 Three-hinged frames 186 Form and stresses 187 Side-by-side frames189 Stacked frames 191 Multistory side-by-side frames 192 Vierendeel beams 192 Deep beams and walls 195 Wall beams.197 Deep wall over several storeys197 Deep walls in space199 folded plate structures 200 Ribbed slabs, beam grids and slabs 201 Composition of beams to support a flat area. 203 Beam grids 204 Slabs 205 One-way slabs 205 Choice of slab thickness 206 Influence of the type of support on slab behavior 208 Continuous slabs 209 functioning with concentrated loads 209 Two-way slabs 209 Equivalent spans for two-way slabs 210 Slabs on columns 211 Mushroom slabs 212 Flat slabs212 Choice of thickness of slabs supported by columns 213 Stability of compressed members 215 Strength of a rod in compression 218 How to make a column stable 219 Influence of column height on compressive strength 220 Influence of the constraints, effective length 221 Influence of the stiffness of the material on the critical load of a column 224 Influence of the dimensions of the section225 Influence of the shape of the section 226 Choice of sections 227 Local buckling227 Trusses and Vierendcel columns. . 228 Variable-section columns228 Appendices 229 Appendix 1. Analytical determination of the funicular curve with distributed load 231 Appendix 2. Analytical expression of the conditions of equilibrium 232