Reader 7P3X0 Structural Design PDF

Table of contents Structural Design Compiled by: Prof. ir. Frans van Herwijnen Ir. Rijk Blok Prof. Ir. Dirk Martens Translation: Sander Montrée Eindhoven University of Technology Faculty: Build Environment Table of contents Table of contents Chapter 1 Introduction 1....

Table of contents Structural Design Compiled by: Prof. ir. Frans van Herwijnen Ir. Rijk Blok Prof. Ir. Dirk Martens Translation: Sander Montrée Eindhoven University of Technology Faculty: Build Environment Table of contents Table of contents Chapter 1 Introduction 1. Preface 2 2. Definition of a structure 3 3. History of construction 4 4. Structural systems 6 5. Structural materials 7 6. Structural Safety 12 7. Loads 15 Loads according to Eurocode 1 18 8. Codes of practice 19 9. The design process 20 10. Difference between designing and analysing 24 11. Example 26 12. Questions chapter 1 28 Chapter 2 Columns 1. History of the column 30 2. Structural meaning 32 3. Stocky and slender columns 36 4. Designing columns 39 5. Column-floor connection 42 6. Column-beam connection 43 a. Timber column-beam connections 43 b. Steel column-beam connections 43 c. Concrete column-beam connections 44 7. Stability of blocks and towers 44 a. Clamped column 45 b. Hinged column 46 8. Global dimensions 46 9. Questions chapter 2 47 Intermezzo: Tilting stability of towers 48  Table of contents Chapter 3 Walls 1. Introduction 50 2. History of the wall 51 2.1 Challenges for structural walls 51 2.2 The Egyptians 51 2.3 The Greeks 51 2.4 The Romans 51 2.5 The Gothic age 52 2.6 The Renaissance 52 2.7 Architecture in the 20th and 21st century 52 3. Construction methods 52 3.1 Construction methods for masonry walls 52 3.1 Construction methods for concrete walls 54 3.1 Construction methods for timber walls 54 4. Masonry strength 56 5. Monolithic and composite walls 57 5.1 The single leaf wall 57 5.2 The cavity wall 57 5.3 The anchorless cavity wall 57 6. Stability of walls 58 6.1 Walls built of infinitely strong and stiff blocks 58 7. Global dimensions 59 8. Questions chapter 3 60 Chapter 4 Arches 1. Introduction 62 2. History of the arch 62 3. Principle of arches 64 Reversibility of tensile and compressive systems. 64 Beam versus Arch 65 Thrust 66 Principle of arch action 67 Rise 69 Arches with symmetrical loading 69 4. Semi-circular arch (Roman arch) 70 Intermezzo: Roman semi-circular arch terminology 71 5. Examples of arch bridges 73 6. Global dimensions 75 7. Questions chapter 4 76 Table of contents Chapter 5 Tensile structures 1. Introduction 78 2. History of tensile structures 78 3. Cable and tension systems 79 4. Force diversion 80 5. Rise 81 6. Flexibility 82 a. Increasing self-weight 83 b. Making a shell 83 c. Stabilizing using an oppositely curved cable. 83 7. Guy cables 84 8. Cable structures 85 9. Questions chapter 5 88 Chapter 6 Beams 1. Introduction 90 2. History of beams 90 Intermezzo: Historical research 91 3. Force distribution 92 Composed beams 95 4. Influence of cross sectional shape 95 Timber beams 98 Steel beams 99 Concrete beams 102 Pre-stressed concrete 103 5. Influence of supports 106 Location of the supports 106 Type of support 107 Number of supports 107 6. Special beams 110 Vierendeel-truss 110 Cable supported beam 111 7. Trusses 112 Intermezzo: Bridge over the River Cismon (1570) 113 Main shapes 114 Intermezzo: Sheldonian Theatre, Oxford (1662-1666) 114 Patterns 114 Structural height 116 8. Example: the design of steel beams or a steel truss 117 9. Guyed structures 120 10. Questions chapter 6 121  Table of contents Chapter 7 Floors 1. Introduction 124 2. History of the floor 124 3. Timber floors 125 Intermezzo: Windsor Guildhall 126 4. Stone floors 127 5. Concrete floors 128 Capacity of concrete floors 130 6. Modern flooring systems 131 Ground floor 131 Story floors 132 7. Example: the design of a floor 135 a. Concrete floor 135 b. Adding columns 135 c. Hollow core slab 136 d. Wide-slab floor 136 8. Questions chapter 7 137 Chapter 8 Portal frames 1. Introduction 140 2. History of the portal frame 140 Intermezzo: Roof beams in farmhouses 141 3. Portal action 142 Portal versus arch 142 Horizontal reaction force 143 Portal versus beam 143 4. Structural Design 145 Three-hinged frames 145 Two-hinged frame 146 5. Connections 148 Steel portal frames 149 Timber portal frames 151 Concrete portal frames 152 6. Questions chapter 8 153 Table of contents Chapter 9 Foundations 1. Introduction 156 2. History of the foundation 156 3. Soil properties 157 4. Geotechnical research 158 Terrain research 158 Laboratory research 159 5. Structural Design 160 6. Shallow foundations 161 Unreinforced strip footing 162 Reinforced rib foundation 162 Reinforced strip footing 162 Pad foundation (or foundation on footings) 163 Basement 163 7. Pile foundations 164 8. Retaining wall 165 9. Groundwater 166 Drainage 167 10. Questions chapter 9 168 Chapter 10 Stability 1. Introduction 170 2. Stability of a horizontal plate 171 a. Clamped base 171 b. Clamped corner 171 c. Stability walls 171 d. Bracing 172 e. Core 172 3. Vertical stability planes 173 a. Truss 174 b. Walls 174 c. Portal frames 174 4. Disc action 175 5. Distribution of wind load 176 a. Façades with horizontal load distribution 177 b. Façades with vertical load distribution 178 6. Stability of an industrial building 179 Longitudinal stability 179 Transverse stability 182 7. Example: stability 184 8. Questions chapter 10 185  Table of contents Appendix A Rules of Thumb for global dimensions 1. Introduction 188 a. Global dimensions by rules of thumb 188 b. Design calculations 188 c. Control calculations 189 Appendix B Literature 214 Appendix C Glossary 220 Table of contents  Chapter 1 - Introduction Chapter 1 Introduction 1 Chapter 1 - Introduction 1. Preface The course Structural Design 1 is the first in a series of courses on designing of structures. From 2013 it forms, in corporation with the course on Applied Mechanics, the new course Analysis of Structures. Structural Design treats, in an integrated manner, the fundamental principles of structural analysis, the application of materials and principles of design. The objective is to gain insight in the force distribution, design and composition of (simple) structures. It also forms the first introduction to the process of structural design. Using the insights and knowledge gained in the course Structural Design 1, the student should have developed adequate tools to create a (simple) structural design during the project work in the 2nd year. After an introduction to the design process, the forces acting on the structure and describing of different structural materials, the structural elements (columns, walls, beams, floors, arches and portals) will be treated. The combination of these elements and their connections will also be addressed. The principles of directing the forces to the ground (foundations) and the stability of low-rise structures will be described and explained by using examples. In order to make safe and economical structures, it is inevitable that a large part of the process of structural design is numerical. Presenting the material, the mathematical or mechanical approach is avoided as much as possible. From the knowledge obtained in lectures on Applied Mechanics and familiarity with the built environment around us, you should develop a natural feeling for structures. In separate chapters framed interludes are included, that give a further explanation of the presented material by giving examples or background information. The text segments that are NOT part of the Examination material have a continuous line in the margin. This also applies to the appendix ‘Rules of thumb for global dimensioning serving the project work in the 2nd year’. 2  Chapter 1 - Introduction 2. Definition of a structure The word Structure can be used to describe an organized system, such as the “structure of management of a company” or the “structure of an atom”. In these lecture notes the definition is as follows: “A structure is a physical system used to direct loads from one place to another” In case of buildings this usually means the safe distribution of loads carried out by people, furniture, wind etc. (and also the self-weight of the building itself) to the foundations, and from there to a (load-bearing) subsurface (see Figure 1.1). From the definition we can conclude, that for structures the following is important: 1. The structural systems used 2. The structural materials that make these systems 3. The forces acting on the structure 1.1 A Structure for buildings will 4. The load-bearing subsurface the structure is placed upon safely transport the forces to The first three topics will be treated in the following paragraphs. For the the earth. 4th topic a separate chapter is included in these lecture notes (Chapter 10: Foundations). In these lecture notes we will restrict ourselves to man-made structures. Nature shows us wonderful examples of efficient structures developed to withstand loads. Some examples are given in Figures 2.1 to 2.3. Although natural structures could also be analysed and researched by using methods applied when designing structures, we are most interested in the exciting process of designing new structures. 2.1 The human skeleton is a structure that maintains the shape of the human body keeps the organs and muscles in place and transports forces to the ground. 2.2 A Spiders web is a good example of a tensile structure. The weight of the spider and its prey is carried by using the 1.2 tensile strength of the wires in the web. 2.1 2.2 2.3 2.3 Shells are extremely efficient structures, they are stiff and lightweight. The curved surfaces can be very thin. 3 Chapter 1 - Introduction 3. History of construction The first man-made structures were pressure-based systems: walls loaded in-plane. The first building materials were timber, natural stone and even bricks. With the exception of timber, these materials are capable of withstanding pressure, but not able to withstand tension stresses. It is thereby logical that structures in old stone-like materials are based on compression. Buildings in timber in the classical ages have disappeared completely, only some buildings in stone have been preserved. The oldest remaining buildings preserved until now are the Dolmen. Another example is the famous Stonehenge, near Salisbury in England. Here we find the first stone beam with a height of 1,5 meter, that spans an opening of 3 meters. In Egypt, timber was a scarce material and the quarries were located at a great distance from the Nile Valley, so stone of sun-dried clay was the only material available for many centuries. The structural systems were extremely rudimentary: load-bearing walls and a flat or slightly sloping roof. The need to span certain areas confronted architects with the invisible gravity, and ways to fight it. Flooring structures using wooden beams, arches and domes originated. Besides a space dividing function, they also fulfilled a structural function, just as the walls of a house. Also elements with a purely structural function arose: pillars. The pillar formed during the Mycenaean civilisation (ca. 1400 BC) on Crete. The Hellenistic temples have pillars made from flat polished marble blocks placed onto each other without mortar joints. The blocks are connected using wooden dowels encased with lead to resist earthquakes (see also chapter 2: Columns). After the architecturally and structurally simple Doric columns, the much more slender Corinthian and Ionic Columns were applied; these were also provided with a pillar base, which had to ensure a smooth force transition to the foundation. On top of the pillars architraves are placed (monolithic stone beams), with a span of ca. 6 meters (e.g. the Acropolis in Athens). The vault technique was known by the Greeks. The Romans on the other hand built bridges, aqua ducts, public baths and basilicas using arches and vaults from concrete, bricks and natural stone. Arches and vaults were the type of structure most used in ancient Rome. 1.5 Temple of Kamak 4  Chapter 1 - Introduction The Gothic style flourished in the 13th century and is widely regarded as the pinnacle of architecture using pressure loaded materials. By an ingenious use of flying buttresses and heavy buttresses they were able to avoid tensile stresses (as a result of horizontal forces in the vaults) in the pillars. Up until the 18th century only timber and stone was used for buildings. The arrival of new materials in the 18th and 19th century led to new structural forms. The development of iron production is closely related to the industrial revolution in England. The iron was more a Structural material than a building material. In 1779, Abraham Darby III and John Wilkinson built the famous cast iron arch bridge over the Severn by Coalbrookdale. This bridge marks the beginning of a period of constructing in iron and later in steel, that continues to this day. The structural material steel has the property that it can withstand both tensile and compressive stresses. Using this material both tensile and compression systems could be developed, which are characterized by a very efficient use of the material. Large spans using tensile systems, such as suspension bridges and cable structures, became possible. The absolute record is the suspension bridge over the bay of Osaka in Japan, with a span of 2 km (Figure 1.6). 1.6 Akashi Kaikyo suspension bridge Mid 19th century the concrete material originally developed by the Romans came back into focus, by the production of Portland cement as a binder for sand and gravel. The downside of concrete, the fact that it is less suitable for tension stresses, was overcome by adding steel reinforcement. This combination of concrete (compression) and steel (tension) gives a fantastic building material to create columns, beams, floors, walls and shells. 5 Chapter 1 - Introduction In summary, in the course of history a shift occurred from the use of compression systems to systems where compression and tension collaborate or only tension is present. This evolution is related to the development of materials that are able to absorb large tensile stresses and the need that exists to make larger spans. 4. Structural systems A structural system is determined by the way it carries the loads. The following systems are distinguished: 1. Structures loaded in compression a] Vertical Columns Loaded according to their axis structures Walls Loaded in plane b] Spanning Arches Loaded according to their axis structures Shells Loaded in plane 2. Structures loaded in tension a] Vertical (tension-) Loaded according to their axis structures columns Cables Loaded according to their axis b] Spanning Cable structure Loaded according to their axis structures Membrane Loaded in plane structures 3. Structures loaded in bending a] Vertical Façade members Loaded perpendicular to their axis structures walls Loaded perpendicular to plane b] Spanning beams Loaded perpendicular to their axis structures Floors or plates Loaded perpendicular to plane 4. Portal frames Assembly of columns and beams using a fixed connection. In the following chapters of Structural systems these lecture notes the Columns and walls different structural systems are discussed. Arches Tensile structures Beams and floors Portals 6  Chapter 1 - Introduction 5. Structural materials The following structural materials are distinguished: Natural materials Stone and Timber Already used as building material for centuries. Characteristics well known by craftsmen. Variable quality / significant defects: careful selection and using high material safety factors (see paragraph: Structural Safety). Artificial produced materials Steel- and Aluminium-alloys Produced under controlled conditions in factories. Production process subjected to various inspections and tests. 1.7 Timber Consistent material: lower material safety factors. Remark: Concrete is an intermediate: artificially made using natural materials: sand, gravel, cement and water. New materials fibre reinforced composites Fully fabricated materials. Around for a short while: behaviour not yet fully understood : high material safety factors for as long as the research is not completed. Old Materials wrought iron and cast iron Used to be applied in structures in the past, now replaced by steel. Below a further description is given of the most applied materials. Steel The basis of steel is iron, to which carbon and other additives are added. Structural steel has a combination of favourable properties: relative strength, cheap, ductile (i.e. not brittle) and the possibility to weld it. Structural steel is available in different types: S235 / S275 / S355 (S stands for steel, the number for the so called yield stress. A type with high strength is used for making rebar for reinforced concrete, referred to as FeB500. An even better type is used for cables and pre-stressed strands: FeP1860. Steel is also used for other products: plates, bars (e.g. rails), profiles (hot- rolled and cold-formed) with standardised dimensions, and wires. 1.8 Detail of a Steel structure 7 Chapter 1 - Introduction Concrete Concrete is made of 4 materials / ingredients: - Cement (e.g. Portland cement or blast furnace cement) - Fine aggregates (sand) - Coarse aggregates (gravel or crushed debris) - Water The ration gravel : sand : cement is about 4 : 2 : 1. Additionally plasticizers are added to increase the processability or other additives to improve the quality (e.g. silicafume for high strength concrete). The strength of concrete is expressed in the characteristic cube 1.9 Concrete structure compressive strength. For example: C20/25 has a characteristic cube compression strength of 25 N/mm². This strength is measured by breaking a concrete cube with edges of 150 mm. The water in the mixture chemically bonds to the cement: hydration. This process normally starts 24 minutes after adding the water. In this stadium we speak of “green concrete”. After 3 to 6 days the concrete has hardened in such a way, that it has reached 2/3 of its final strength. Concrete is a brittle material with a tensile strength of about 10% of its compressive strength. Therefore concrete is strengthened with rebar to take the tensile stresses. We call this “reinforced concrete”. Timber In timber we distinguish 2 types: - Hardwood (deciduous) - Softwood (coniferous) Remark: The terms hardwood and softwood are somewhat misleading. Balsawood is classified as “hardwood” but is one of the softest woods. Conifers grow faster than deciduous trees, making softwood cheaper than hardwood. Structures are therefore mostly made using softwood. With regard to the mechanical properties, timber is classified in the following strength classes: C14 / C16 / C18 / C22 / C24 / C27 / C30 / C35 / C40 / C45 and C50. C stands for coniferous and the number 1.10 Timber structure represents the characteristic bending strength parallel to the grain, e.g. C18 has a characteristic bending strength of 18 N/mm². Timber is an anisotropic material, meaning that the properties in different directions (i.e. parallel and perpendicular to the grain) vary. In timber different products are made: planks, beams, bars and plates. 8  Chapter 1 - Introduction Masonry Structural masonry includes building with bricks, natural stone blocks, concrete blocks, Calcium Silicate bricks, cellular concrete etc. All these are brittle materials with a low tension strengths, mostly used in structures loaded in compression. An exception is made for freestanding (garden) walls loaded by wind. In these walls bending moments occur and it is assumed the masonry can take the minor tension stresses. Bricks are made of clay mixed with water, shaped and baked in ovens. Bricks are made in standardized sizes (Waal-size, Maas-size). Bricks can be stacked with mortar to make brick masonry. The properties of the 1.11 Masonry structure masonry are different from the properties of the individual stones. Natural stone is a relative expensive building material. The strengths vary depending on the type of stone. There are no standardizes sizes. Concrete blocks are one of the cheapest building materials. They are made as solid or hollow blocks (to reduce weight). Concrete blocks are available in standardized sizes. Calcium Silicate blocks are made by pouring a mixture of sand, cement, lime and water into moulds and placing them in an autoclave (oven), where they are cured by high pressure and temperatures. Masonry structures are discussed in more detail in chapter 3 (Walls). Aluminium Aluminium is made from bauxite. It is a lightweight, strong and corrosion resisting metal that in some cases is a good alternative to steel. The production process is energy consuming; the reason an aluminium product is almost twice as expensive per unit of volume as steel. Pure aluminium is too soft and weak to use in structures, but by adding 5% other materials, such as magnesium and silicon, the properties are enhanced to make a suitable structural material. By using aluminium alloys you should take into account: - The low modulus of elasticity (about 1/3 of steel) gives a large 1.12 Aluminium Structure elastic deformation; - The maximum allowable temperature, where the strength doesn’t decrease: 125 to 200°C. 9  Chapter 1 - Introduction Prestressed concrete Reinforced concrete Material properties Aluminium Masonry C20/25 C35/45 Timber Steel S235 C16 Characteristic 𝜎𝑑 N/mm² 3 9,9 235 15 27 125 compression strength Characteristic tension 𝜎𝑡 N/mm² - 9,9 235 - - 125 strength Modulus of elasticity E N/mm² 5.000 10.000 210.000 28.500 33.500 70.000 Coefficient of expansion 𝛼 C-1 0,5·10-5 0,5·10-5 10-5 10-5 10-5 2,4·10-5 Density 𝜌 kg/m3 1.800 550 7850 2.400 2.500 2.600 compression strength - m-1 167 1.800 2.994 625 1.080 4.808 own weight Cost - €/m3 400 400 9.000 500 600 18.000 Cost - - 0,134 0,041 0,039 0,034 0,022 0,144 compression strength Tabel 1.1 Material properties for the most important structural materials. Fibre reinforced composites Fibre reinforced composites are composed of two distinct materials: fibres to take the tensile stresses and a resin as a matrix. Fibres used are: As matrix used: - Glass fibres - Polyester - Carbon fibres - Vinyl ester - Epoxy Table 1.1 gives some characteristic properties of the most important structural materials. For the material properties, average values are given for the conventional types. The given costs are an indication, the real costs are depending on the quantity, the complexity of the structure and market values. An ideal structural material is lightweight, so the self-weight is small compared to the total load acting on it. 10  Chapter 1 - Introduction The stress-weight-ratio (SWR) is calculated by taking the characteristic strength (in kN/m²) and dividing it by the volumetric weight (in kN/m³). using the SWR you can determine the maximum allowable length of material if an element is only loaded by its own weight. Remark: As we will see in paragraph 7: Loadings, the self-weight will be increased with a factor 1,2 during calculations. Example: 1860 ∙ 106 Steel cable FeP1860 𝑆𝐺𝑅 = = 19.745 𝑚1 78500 ∙ 1,2 Masonry 3000 𝜎𝑑 = 3000 𝑘𝑁/𝑚² 𝑆𝐺𝑅 = = 139 𝑚1 tower 18 ∙ 1,2 Properties of structural materials Strength and stiffness are probably the most important properties when we have to decide if a material is suitable to use in structures. The strength of the material determines the ultimate limit state (ULS) of the structure (the load at which the structure collapses). The stiffness of the material determines the serviceability limit state (SLS) of the structure (for slender columns loaded in compression this is equal to the Tensile force Ultimate Limit State!). The strength and stiffness can be determined by one single test, for instance a tensile test on a steel bar. During testing, a graph can be drawn where the stresses and strains are plotted (see Figure Cross section 1.14). tensile force stress σ= cross section elongation Strain ε= 1.13 Scheme for a tensile test original length 𝑠𝑡𝑟𝑒𝑠𝑠 Modulus of elasticity E = tan ( ) 𝑠𝑡𝑟𝑎𝑖𝑛 𝜎 Furthermore the following properties are important: - Sustainability - Fire behaviour - Fatigue - Density - Toughness / brittleness - Costs - Creep behaviour - Environmental impact Arctan E 𝜀 1.14 Stress-strain relationship for an bar loaded in tension 11 Chapter 1 - Introduction 6. Structural Safety *) *) the Master-course “Structural Design 6; Safety, Reliability and Loadings” covers this topic in more detail. In this section a general When designing a structure, the structural designer should prove that the overview is given. loads acting on the structure are smaller than the resistance of the structure. The term ‘loads’ also includes imposed deformations, expansion due to changes in temperature, and shrinkage. The effect of the loads on the structure depend on a number of variables, of which the nature of the load, the dimensions of the structure, and the material properties are the most important ones. Using mechanics the effect of the loads can be expressed in terms of internal moments and forces, normal- and shear forces, and the accompanying deformation. By doing this the structural engineer will gain insight into the required dimensions of the structure. After the dimensions of a structure are set, the cross-sections can be determined. The values of loads and material properties can be found in the codes of practice. The process of assessing the structure follows a deterministic path. The structural engineer knows 1.15 Collapsed structure that the variables in real life don’t have the exact value as assumed for the calculation. This is because the variables are in fact statistically stochastic **) variables: the loadings vary in time, dimensions are between limits of **) stochastic: depending on chance. tolerance and material properties are scattered over a certain range. The engineer should therefore show that the chance of failure of the structure is sufficiently small. In the Netherlands the acceptable probability of failure is set at 10-6. Determining that the chance of failure of the structure is sufficiently small is achieved by taking a characteristic value for the loadings ***) and multiplying these values by a loading factor; and for the ultimate limit ***) see paragraph 7: Loadings state determining a characteristic strength and dividing this by a partial safety factor (material factor). The desired chance of failure is not exceeded when it is proven that the effect of the characteristic loads, multiplied by a loading factor, is not larger than the characteristic strength, divided by the partial safety factor (see Figure 1.16) 1.16 Graph showing probability density functions for the strength and loads. F = Probability density function S = Solicitation of stress R = Capacity of resistance Both S and R are stochastic variables 12  Chapter 1 - Introduction The characteristic values of the loads will not be exceeded in 95% of cases. The real values of the strength will not be under the characteristic values in 95% of cases. Both the loads and the strength of materials are recorded in codes of practice. To summarize: The design load 𝑆𝑑 is found by multiplying the characteristic load 𝑆𝑘 by the load factor 𝛾𝑠. The design strength 𝑅𝑑 is found by dividing the characteristic strength 𝑅𝑘 by the partial safety factor 𝛾𝑀. The structure is sufficiently safe if it meets the requirements: 𝑅𝑘 𝑆𝑘 ∙ 𝛾𝑠 ≤ 𝑆𝑑 ≤ 𝑅𝑑 𝛾𝑀 Design strength The design strength of a structure follows from the strength function R of the design model. This model is based on theoretical considerations and on observations of real behaviour of structures during tests. For example: the strength function for a steel bar loaded in tension is: 𝑅 = 𝐴 ∙ 𝑓𝑦 In this formula, A is the cross section of the bar, and 𝑓𝑦 the yield stress of the steel. The suitability of the chosen design model can be reviewed by comparing the results of the strength function to test results of bars loaded in tension. The design model is modified until the theoretical values and the test results are sufficiently corresponding to each other. When this is the case, the strength function R is adjusted and the characteristic strength Rk is formed, expressed in nominal values for the dimensions and design values for the material properties. Finally the partial safety factor (the material factor) 𝛾𝑀 is determined and the design strength follows: 𝑅𝑘 𝑅𝑑 = 𝛾𝑀 The material factors for structural materials are: - Steel 𝛾𝑀 = 1,00 - Timber 𝛾𝑀 = 1,20 - Concrete 𝛾𝑀 = 1,40 - Stone 𝛾𝑀 = 1,80 13 Chapter 1 - Introduction Design load Not only the strength, but also the loads are important when assessing the safety of a structure. However, the load is not described as a characteristic value combined with a certain probability of exceeding. There are different kinds of loads. Also the load to be assessed depends on the location of the structural element. For a beam supporting a roof the snow load should be taken into account in the load combination. For the same beam supporting a floor the snow load is no longer important, but an expected floor load should be applied in the load combination. The following paragraph will treat this in more detail. Safety From an economic point of view it is not useful that every building has the same structural safety. For example, we desire a larger safety for nuclear power plants than for a greenhouse. The criterion for the definition of safety is the probability of failure, multiplied by the damage caused in case of collapse. This also includes the risk to human lives. 1.17 Collapsed building in San Francisco. Reliability classes ( remark: reliability classes = Consequence Classes) It is impractical, and impossible, to determine the desired structural safety for every type of building. This is why the codes of practice use a division of three reliability classes. Depending on the use of a structure a minimum requirement is given. It is always permitted to choose a higher reliability class. The partial safety factors are related to the reliability classes. Reliability class 1 This class applies to structures in which no people are present during extremely bad weather. If collapse occurs the damage is restricted to the structure itself and present goods. 1.18 Greenhouse Example of class 1 structures: greenhouses. Reliability class 3 This class applies to structures that will not only give great financial impact if collapsed, but also large human, emotional or social damage. Buildings that house basic services or functions that cannot be missed by society also apply to this class. Example of class 3 structures: residential buildings, office buildings, bridges and power plants. Reliability class 2 This class applies to all structures not in classes 1 or 3. Example of class 2 structure: single family houses. 1.19 Office building 14  Chapter 1 - Introduction 7. Loads Structures are made to direct loads from one place to another. There are different types of loads that can act on a structure. For instance, take a residential building consisting of floors, loadbearing walls and a roof. First there is the self-weight of the structure itself and the elements permanently attached to it. Then there are variable loads working on the structure, think about people and furniture in the house, snow on the roof during winter, and pressure and suction from wind on the roof and façade. Because these loads vary in time and the size is difficult to predict exactly, values have been established by experts, which can be used in common cases. These values are written down in standards in the codes of practice, so every structural engineer uses the same principles for determining the loads of the floors, roof and façade of buildings. In Europe the Eurocode is used. Eurocode 1 (EN 1991) describes all load cases that should be taken into account. Eurocode The Eurocode consists of 10 volumes. The first (Eurocode 0) is general for the basis of structural design. Eurocode 1 is about actions on structures. Eurocodes 2 to 6 and 9 are material specific. Eurocode 7 is on foundations and Eurocode 8 is on Earthquake design. EN 1990: Basis of structural design EN 1991: (Eurocode 1) Actions on structures EN 1992: (Eurocode 2) Design of concrete structures EN 1993: (Eurocode 3) Design of steel structures EN 1994: (Eurocode 4) Design of composite steel - concrete structures EN 1995: (Eurocode 5) Design of timber structures EN 1996: (Eurocode 6) Design of masonry structures 1.20 Eurocode 1: Actions on EN 1997: (Eurocode 7) Geotechnical design Structures EN 1998: (Eurocode 8) Design of structures for earthquake resistance EN 1999: (Eurocode 9) Design of aluminium structures 15 Chapter 1 - Introduction Types of loading Eurocode 1 names three types of loading: - permanent loads - variable (live) loads - accidental loads Permanent loads are loads that vary little in time. The most important permanent loads are the self-weight of the building components, ground pressure for basement walls, loads by fluids in storage tanks and pre- stress. Variable (live) loads vary in time and size. For instance wind: sometimes there is almost no wind, most times there is little wind, and once in a while there is a storm. Live loads can be anything from people, furniture, decoration of spaces, storage of goods and materials, loads from machines and vehicles and influences from the weather such as rain, wind and snow. Accidental loads are characterized by large and disastrous consequences, and a low probability of occurrence. These loads occur for instance during (gas)explosions, fire, collisions of vehicles and earthquakes. The different types of loads have different characteristics. In particular the probability of occurrence at any given point in time differs significantly (see table 1.2). Consequently, there are two major questions of which Eurocode 1 provides the answer: - What is the characteristic value of the live load? Should the calculation include a thick layer of snow or only about 10 cm? - Which loads should be taken into account in a load combination and to what extent? Permanent loads are always present, live loads sometimes and special loads (almost) never. Table 1.2 types of load and the Type of load Indication Chance of occurrence chance of occurrence. Permanent load 𝐺 Always instantaneous 𝑄𝑚 Almost always Variable (live) load Extreme load 𝑄𝑟𝑒𝑝 rarely Accidental load 𝐹𝑎 (almost) never 16  Chapter 1 - Introduction Characteristic value of live loads The live load varies in size: we know this from experience and it has been verified by measurements. The wind load for instance is derived from the speed of the wind, and the size and shape of the building. In the Netherlands the speed of the wind is measured every 10 minutes in a number of meteorological stations. From these measurements a mean speed and standard deviation is determined. The characteristic value has a chance of exceeding of 5%. The wind speed at a random time follows from the statistical distribution. This value is called the instantaneous or quasi-permanent value of the wind and is about 11 m/s. the instantaneous value as a measurement for the wind speed at a random moment in time, but in storms this value is exceeded. That is why not only the wind speed at a random time is important, but also the largest wind speed expected to act on the structure in its lifetime. The lifetime of a structure is a difficult concept: is it the total lifetime or the economic lifetime? That is why the Eurocode defines a reference period: a time period in which the structure must fulfil the requirements and the maximum variable loading is expected. To determine the size of the variable load the Eurocode uses a reference period of 50 years. This means that the maximum load occurs once every 1.21 Deformation of palm trees 50 years. This load is called the extreme value of the variable load. On the due to wind from the right basis of wind measurements every 10 minutes it is possible to estimate side. the extreme wind speed. For each period of 50 years the maximum speed is determined from the measurements. Thereby creating a new group of observations, from which the maximum speed is determined with an accompanying standard deviation. From this data the extreme value of the wind speed can be calculated. For the Netherlands the maximum wind speed is about 27 m/s. The ratio between the instantaneous value and the extreme value is given by Ψ (psi). Eurocode gives values for the loads on floors and roofs, assuming both a uniformly distributed load and a concentrated point load on a random location on the floor or roof (see Table 1.3). 17 Chapter 1 - Introduction Table 1.3 Loads according to Eurocode 1 Loads according to Eurocode 1 𝑄𝑚 𝑄𝑟𝑒𝑝 Ψ [kN/m²] [kN] Floors Dwellings / houses 1,75 3,00 0,4 Office 3,00 3,00 0,5 Store 4,00 7,00 0,4 Stations 5,00 7,00 0,25 Industrial hall > 6,00 10,00 0,8 Roofs roofs 1,00 1,50 0 Example: A floor of an office building: Extreme value of variable (live) load 3,0 kN/m² Quasi-permanent variable (live) load 0,5 ∙ 3,0 = 1,5 kN/m² Point load 3 kN Working on an area of 0,5 x 0,5 m, on most disadvantageous location. Design load By multiplying the characteristic value of the loads with the load factor 𝛾𝑠 the design value of the load is obtained. For reliability class 3 the load factors are: - Permanent loads 𝛾𝑠;𝑔 = 1,2 - Variable loads 𝛾𝑠;𝑣 = 1,5 - Accidental loads 𝛾𝑠;𝑎 = 1,0 Variable loads have a larger uncertainty than permanent loads, the reason why a larger load factor is used. Accidental loads are so rare a safety margin is not required. 18  Chapter 1 - Introduction Load combinations To test the safety of a structure the Ultimate Limit State (ULS) and the occurring loads are determined. Permanent loads (G) are always applied, and are therefore present in all load combinations. Quasi permanent loads of the variable load (𝑄𝑚 ) are also expected to be present at all times, and also present in all load combinations. Extreme values of the variable loads (𝑄𝑟𝑒𝑝 ) and accidental loads (𝐹𝑎 ) occur seldom. Statistically it is highly unlikely that two types of (independent) variable loads (e.g. maximum snow load on a roof and a maximum floor load on a story) simultaneous reach their extreme values. This means that this load combination does not have to be taken into account. The chance of the accidental load acting simultaneous with the extreme variable load (𝐹𝑎 + 𝑄𝑟𝑒𝑝 ) is also negligible. Thus, we arrive at the following two load combinations in accordance to the rules of Eurocode 1: 1) Fundamental combinations Permanent load 𝐺 ∙ 𝛾𝑠;𝑔 One extreme variable load 𝑄𝑟𝑒𝑝 ∙ 𝛾𝑠;𝑣 Other quasi permanent loads ∑ 𝑄𝑚,𝑖 ∙ 𝛾𝑠;𝑣 2) Accidental combinations Permanent load 𝐺 ∙ 𝛾𝑠;𝑔 Quasi permanent loads ∑ 𝑄𝑚,𝑖 ∙ 𝛾𝑠;𝑣 Accidental load 𝐹𝑎 ∙ 𝛾𝑠;𝑎 8. Codes of practice Calculations of structures are to be addressed following normalized values of loads and material strengths. These values can be found in a combination of codes: the Eurocodes (see previous description). In Eurocode o the requirements (reliability and serviceability) are given at a fundamental level, to which all building structures must comply, regardless of the material they are made of. These fundamental requirements are worked out in more detail to the aspects in Eurocode 1 (determination of loads and resistance of structures) and for the structural materials. 19 Chapter 1 - Introduction 9. The design process The best way to explain the general approach followed by an engineer solving a structural problem, is by using a simple example: the design of a swing. It is obvious that for such a small project the formal procedure described will not be followed consequently. The designer will determine the structure and size of the elements to his own insight. Nevertheless, it gives an example of the steps involved to successfully treat larger projects. A common starting point of a project is a client with a certain problem initiative design that approaches a designer, often an architect or engineer, and gives the assignment to make a design to fulfil the needs. The first thing a designer will ask, is a detailed description of what the client wants; a so called Program (or Program of requirements). Example the Client is a child that wants a swing. project definition / The program is: feasibility “Build a swing that can be played with for many years in two weeks.” To arrive at a satisfactory solution, the following steps will be taken: Phase 1: Research of the location Suitable locations are investigated and checked to make sure no obstacles are in range of the swing. The accessibility of the building site should be project definition / checked out, and the quality of the soil during rain should be taken into feasibility consideration. Furthermore some test holes should be dug, to check for any debris close to the surface of the soil which could make excavations for the foundations difficult. 20  Chapter 1 - Introduction Phase 2: Designing alternative structures sketch design There are numerous structural shapes and materials that can be used. Some of them are shown in Figure 1.22. Each alternative should now be evaluated on clear rational criteria, in order to select one or two solutions for further research. The factors considered for the alternatives are: a) Connected to a tree; this is the cheapest and best option, but (luckily) no suitable tree is present. b) A frame of steel tubes; available in DIY-shops. Little effort is needed and the swing can easily be moved. Some concerns on costs, durability and appearance can come to mind, but an option worth considering. c) Hung from a helium balloon; interesting option, but very impractical: not fixed in place. d) Steel beams with dowel or welded connections; very robust, but can be prefabricated by a local steel company. The result is exceptionally hideous and requires an additional coating to prevent corrosion. e) Timber frame with concrete foundation; a nice appearance and relatively cheap, but a lot of work required and maintenance required for durability. f) Timber and steel console attached to wall; although it is possible to fix the swing away from the corner, safety can still be an issue. The alternatives worth investigating are (b) and (e). we have to assume that after careful consideration of available products the final choice fell on the timber frame. a b c d e f 1.22 Design alternatives 21 Chapter 1 - Introduction Phase 3: preliminary design preliminary design The preliminary design of the structure involves the selected structural scheme. The basic dimensions should be determined. A braced frame on behalf of the stability of the portal is needed. In this phase also the type of timber (strength class) and the available cross sections are chosen. Phase 4: determining loads preliminary design This might not be as straight forward as one might expect. The following topics are in order: 1. Does the swing have to be designed only for children, or should we take into account that adults also sit on it during the lifetime of the swing? My suggestion is to certainly take this into account! 2. We have to increase the load from the weight of a child due to a dynamic load. This means that the load moves and we also have to introduce the effects of inertia of the mass into account. Also the load can act suddenly if one jumps onto the swing. A “dynamic enlargement factor” f=2 should take care of these effects. 3. The load doesn’t always act vertically. When the swing is in motion the structure will be loaded in bending and by horizontal forces. Also wind gives a horizontal load, although this is minor considering we are dealing with a rod-like open structure. Remark: This is not always the case, think for example about special trusses with bars in different planes. 4. Additional loads due to the self-weight of the structure. When 1.23 Preliminary design dealing with large bridges the weight of the bridge is normally larger than the loads acting on the bridge by traffic. In this case, the load of the structure forms no problem, because we don’t know the dimensions (cross sections) of the beams (and thereby the weight). The engineer should make a reasonable guess, that has to be verified later. It is clear that an experienced engineer will be able to make these kinds of estimates more accurate. In this case the effect of the self-weight of the structure are very little, and therefore negligible. Remark: in the annexes of these lecture notes, rules of thumb are given to make these first assumptions. 22  Chapter 1 - Introduction final design Phase 5: calculation of the structure First the structural drawings have to be translated into a simple scheme (we call this schematizing) to apply the structural mechanics. This meanly involves classifying of the connections between members as hinged or fixed connections. In general, fixed connections in timber are not easily made, therefore we assume hinged connections. The final mechanical scheme is shown in Figure 1.24. The loads are now shown and the portal frame is analysed to find the deflections, the normal (axial) forces and shear forces and the bending moments. It will be needed to make several calculations for different load combinations. For this swing there are two load combinations: - only vertical loadings - swing in extreme deflection Phase 6: Design of elements and details Every element should be reviewed separately. Taking the forces and deflections into account as calculated in the analysis of the total structure, the dimensions of the individual members have to be determined in such a way that the calculated values do not exceed the strength. Special care should be taken for the connections, so that the forces can be transferred from one member to another. The structural engineer has to evaluate all possible ways the structure could collapse. Some of the possibilities for the swing are given in 1.24 Mechanical scheme of the Figures 1.25 and 1.26, other failure mechanisms are also possible. swing 1.25 Faillure mechanisms a. Failure of the seat b. Failure of the rope c. Horizontal slanting d. Upper member fails a b c d 23 Chapter 1 - Introduction 1.26 Horizontal slanting and two possible solution to prevent it. After the final design the phases beneath follow: technical design - more detailed elaboration - contractors calculate their price and present their offer Problem method 1 method 2 execution design - all elements are finalized for production. 10. Difference between designing and analysing execution - the building is created on site Analysing The term analysing is generally used to refer to the process of calculating a certain structure with known loads in order to find the distribution of forces in the various elements of which the structure is constructed. This also includes the determination of the stress distributions in the individual elements resulting from the loads acting on the elements. Finally, the calculation also includes the determination of deformations caused by a certain combination of loads. The analysis of a structure is required to prove that the structure is strong enough to carry a given load. Calculations are based on the laws of applied mechanics, and the goal is to get as close to the real value as possible. Calculations are a vital part of the design of a safe and economic structure; however, they are generally not able to be carried out until the basic shape of the structure is determined. We first need answer questions such as “should we use steel or concrete?” and “how many supports will we apply for that beam?”. These early decisions have to be made in the phase of design, not during the analysis. Design Design is more difficult to define than analysis, because it can have different meanings for different people. The term designer is generally applied to address designers of e.g. patterns on fabrics or the shape of a car’s bodywork. Although aesthetically design is important, it is not the scope of this course. 24  Chapter 1 - Introduction Structural Design, which is the scope of this course, is as much of a creative process; think of the beautiful designs for the Forth Railway Bridge (Figure 1.27 a) and the Golden Gate Bridge (Figure 1.27 b). Two very distinct solutions for the same problem: making a large span over a river. 1.27 The Forth Railway Bridge (a) and the Golden Gate Bridge (b) a b Also in our profession the term design can have two meanings. The first is to describe the entire creative process of finding a safe and efficient solution for a question. Consider the design of a bridge over a river. There is not a single true answer, but from the numerous solutions some are clearly better than others. It is the task of the designer to find the best solution, given the available opportunities and boundary conditions. The second meaning is more restricted. It refers to the work after the phase of analysis, when the (internal) forces in each member are known. It is the process of determining the dimensions of a certain steel column or the amount of rebar in a concrete beam. This is called the design of members. The design of the connections, of which the distribution of forces is clearly visible, forms one of the important points of focus for a structural engineer. The quality of structures is often determined by the details. The design also has to quantify the connection; determining the number of bolts or the length of the weld. This is called designing of details. Both are very important and a badly designed detail is not seldom the cause of major structural damage! 25 Chapter 1 - Introduction 11. Example An industrial hall with a width of 14 meters is spanned by rafters in cross direction (L1 - L4) and supported by columns in the façade (K1 - K4) with a centre-to-centre distance of 6 meters. The rafter is a steel profile IPE 450, and has a self-weight of 0,78 kN/m. On top of the rafters an isolated steel roof is designed. The corrugated steel plates span from rafter to rafter. The live load on the roof is, according to the Eurocode, 1,0 kN/m². Asked: a. Determine the permanent load on rafter L2. b. Determine the variable load on rafter L2. c. Determine the total load on rafter L2. 26  Chapter 1 - Introduction Answer a. The permanent load on rafter L2 is composed of the self-weight of the rafter and the weight of the steel roofing, isolation and roofing material. First, we determine the self-weight of the roof: Steel sheets (from manufacturer) = 0,15 kN/m² 120 mm isolation 0,12 ∙ 0,25 = 0,03 kN/m² 2-layer bituminous roofing = 0,07 kN/m² Total = 0,25 kN/m² The imposed load from this roof on the rafters, expressed in kN/m, is determined by multiplying the weight of the roof, in kN/m², by the width of the roof carried by one rafter. This is equal to the centre-to-centre distance. Imposed load 0,25 𝑘𝑁/𝑚2 ∙ 6 𝑚 = 1,50 kN/m Own weight rafter = 0,78 kN/m Total = 2,28 kN/m b. The variable load on the rafter, expressed in kN/m, is determined by multiplying the variable (live) load on the roof, in kN/m², by the width of the roof carried by one rafter: the centre-to-centre distance. Live load 1,0 𝑘𝑁/𝑚2 ∙ 6 𝑚 = 6,00 kN/m c. The total load on rafter L2 equals: Permanent load 𝑔𝑟𝑒𝑝 = 2,28 kN/m Live load 𝑞𝑟𝑒𝑝 = 6,00 kN/m Total = 7,28 kN/m Remark For the analysis of the strength of the rafter the loads should be increased for safety. The design value of the loads is obtained by multiplying the representative values by load factors. When this industrial hall is categorized as Reliability class 2 the calculation values become: Permanent load rafter 𝒈𝒅 = 𝒈𝒓𝒆𝒑 ∙ 𝜸𝒈 = 𝟐, 𝟐𝟖 ∙ 𝟏, 𝟐 = 2,74 kN/m Variable (live) load rafter 𝒒𝒅 = 𝒒𝒓𝒆𝒑 ∙ 𝜸𝒒 = 𝟔, 𝟎𝟎 ∙ 𝟏, 𝟓 = 9,00 kN/m Total load rafter 𝒒𝒕 = 𝒈𝒅 + 𝒒𝒅 = 𝟐, 𝟕𝟒 + 𝟗, 𝟎𝟎 = 11,74 kN/m 27 Chapter 1 - Introduction 12. Questions chapter 1 1. What do we mean by a structure? 2. What is the difference between the design and analysis of structures? 3. Describe, in words, the meaning of the formula 𝑆𝑑 ≤ 𝑅𝑑. 4. Which types of loading does Eurocode distinguish? 5. Give examples of the types of loads from question 4. 6. What is the difference between the quasi-permanent and the extreme value of a load? 7. What considerations act on determining the required safety of structures? 8. How do we call the codes of practice, in which the size of loads, maximum allowable deformations and methods of determining the resistance of structures are described? 9. What is the difference between the characteristic value of the material strength and the design strength? 10. Which structural materials are used in the Netherlands? 11. What is the stress-weight-ratio (SWR)? 12. Compare the SWR of timber (K17) and steel (S235), if the strength of the timber is 12 N/mm². 13. Which properties of structural materials are most important for a successful application in structures? 28  Chapter 2 - Columns Chapter 2 Columns 29 Chapter 2 - Columns 1. History of the column Columns are among the oldest and most elementary building components in the built environment. In historical buildings we normally speak of pillars; columns with a nearly perfect circular cross- section. Besides a formal and symbolic function, the pillar mostly fulfils a structural function in the various cultural periods: transferring concentrated vertical loads. The pillars in Greek architecture can be distinguished by shape into three categories: Doric, Ionic and Corinthian. The Greek pillar consisted of marble “discs” (weighing up to 6 tons), stacked on top of each other. Therefore the edges were flattened, so that the compressive forces could be transferred there. To prevent the horizontal sliding of the discs, they were fixed using wooden thorns through the heart (see Figure 2.1). The pillar shapes in Roman architecture were based on the three Greek styles, complimented with three styles of their own: Tuscan, Doric- Roman and Composite. 2.1 The structure of a Doric pillar. 2.2 6 pillar styles a. Doric b. Ionic c. Corinthian d. Tuscan e. Doric-Roman f. Composite a b c d e f 30  Chapter 2 - Columns The Egyptian pillars of the New Age were a stone version of the bound papyrus stems, as used for the construction of temporary housing. In the transition into stone, the delicate points of the leaves gain the strength to carry the lintel. Originally the pillars were painted in a variety of colours. The “nature turned into stone” was only applied to buildings that had to be durable, buildings devoted to gods and pharaohs. The palm-capital (Figure 2.4) is used until the Roman Empire. The composite-capital was exceptionally beautiful; the bundle of plants, formed into a cylindrical pillar shaft, unfolds to flowers and leaves in the capital (top of the pillar) (Figure 2.5). 2.3 The bundled papyrus and the stone pillars 2.4 Palm capital. 2.5 Composite capital. 2.4 2.5 2.6 Different types of capitals. 31 Chapter 2 - Columns Columns can be considered as the result of concentrating the vertical load on a wall into a number of small sections (the columns), and removing the parts of the wall without vertical load (see Figure 2.7). 2.7 The creation of the Column. Louis I. Khan: “O what a wonderful day when the wall parted and the column was born”. block walls columns In the Gothic architecture, we see the first clear application of this principle to achieve a maximum entry of daylight through the walls of the nave of the church. 2. Structural meaning Columns are (nearly) vertical structural elements, which are primarily loaded in compression, but sometimes in tension. It is also possible for columns to be loaded in bending (for instance by wind load in combination with over- and/or under-pressure, when applied in façades). When the ends of a column are connected to other structural elements (floors, beams etc.) with a fixed connection, moments can 2.8 Various loadings: from left to occur in the column. These right: compression, tension, moments can also lead to bending compression and bending. of the column. 2.9 Pendulum column 32  Chapter 2 - Columns Pendulum columns have hinged connections, thereby the column is loaded centrically, which means that the line of thrust coincides with the longitudinal axis of the column. For a combination of horizontal and vertical loads on top of the column, the line of thrust is in the direction of the combined force (see Figure 2.10). This is the case in gothic cathedrals (Figure 2.11). 2.10 Lines of thrust in columns. centerline aslijn centerline aslijn compression compression line druklijn line druklijn Line of thrust equals the Line of thrust is unequal to the longitudinal axis longitudinal axis 2.11 Right: cross section of a gothic cathedral. Left and middle: scheme of the distribution of loads from the vault. 33 Chapter 2 - Columns The ribbed vaults of the nave are carried by the pillars of the nave. However, the forces from the vault are not vertical, but directed outward (force A in Figure 2.11). This force can be decomposed into a vertical force B (vertical, to be taken by the column) and a force C (oblique, to be taken by the flying buttress). The force C exerts an oblique, outward directed force on the buttress. The mass on the pinnacle on top of the buttress (D) and the mass of the buttress itself (E) compel the force C downward, so that the line of action of the resulting force (R) of C and D+E stays inside the section of the buttress, and no tensile forces occur on the inside of the buttress. a Remark: The buttress also has to withstand shear and overturning a forces (Figure 2.12). b b 2.12 Failure due to a. Cracks from bending at the base b. Shear c. Toppling 2.13 Buttress with pinnacle. 34  Chapter 2 - Columns For eccentric placement of the normal force P in columns of stone-like material (masonry, unreinforced concrete, natural stone etc.), the line of action should stay within the so called core section (Figure 2.14); this means that the 𝒆 ≤ 𝟏𝟔 ∙ 𝑩, in which e is the eccentricity relative to the column axis, and B the width of the column. By doing so, no tensile forces will occur in the cross section of the column, so the risk of cracks is avoided. Stone-like materials are barely capable of taking tensile forces. 2.14 Centric and eccentric loaded column. Kerndoorsnede gearceerd 35 Chapter 2 - Columns 3. Stocky and slender columns F A short, so-called stocky column is a column that has a large cross section (A) compared to its height (𝓵). For instance, a rectangular column is “stocky” if 𝓵/𝒅 ≤ 𝟏𝟓 when F horizontally supported at the top, or 𝓵/𝒅 ≤ 𝟏𝟎 if not supported at the top, L where d is the smallest width of the cross section (Figure 2.15) For a centric load F the stress equals 𝝈𝒄 = 𝑭/𝑨 [N/mm²]. The column L fails in compression when 𝝈𝒄 > 𝒇𝒄 where 𝒇𝒄 is the compressive strength of the material. This strength is independent of the shape of the cross section. Example A stocky column is loaded by a compressive force of 1260 kN. The design >d value of the compressive strength of concrete is 15 N/mm². What are the d d required dimensions of the cross section, in order to avoid failure? 2.15 A stocky column Answer 𝟏𝟐𝟔𝟎∙𝟏𝟎𝟑 The design value of the compressive strength is: 𝝈𝒄 =. This value 𝑨 𝟏𝟐𝟔𝟎∙𝟏𝟎𝟑 should be lower than the compressive strength: 𝝈𝒄 = 𝑨 ≤ 𝟏𝟓. Therefore 𝑨 = 𝟖𝟒. 𝟎𝟎𝟎 mm2. A square column of 300 x 300 mm (A=90.000 mm² ) is sufficient. In the architecture up until the 18th century, mostly heavy stone columns were used, that behaved as stocky columns. With the first application of cast iron and steel columns, the dimensions were reduced in such a way that the buckling phenomenon occurred (Figure 2.16). In the Bibliothèque Nationale in Paris (1861), designed by Henry Buckling Labrouste (see Figure 2.17), we clearly see the large differences in slenderness of the cast iron columns on the foreground, and the stone columns in the background. 2.16 A slender column can buckle 2.17 Bibliothèque Nationale, Paris, 1861. Arch. Henry Labrouste 36  Chapter 2 - Columns Buckling is the phenomenon that a slender column, which is increasingly loaded in compression, suddenly deflects sideways at a certain load. If, for elements loaded in compression, the compressive force would be exactly in the centre of the cross section, under all load conditions, no instability problems would occur. However: - Straight bars are never exactly straight (geometrical imperfections in for instance timber members); - The external forces are likely to have a minor eccentricity; - Often a lateral load is present that gives the bar a deflection, then the line of thrust and the members axis do not coincide (for instance for columns in the façade). It took a long time before they were able to describe the buckling problem mathematically, and could calculate the load at which the column buckles, the so-called buckling force. Leonardo da Vinci (1452 - 1519) was the first that used the knowledge on statics to determine the forces in separate structural elements. In fact, he was the first that tested structural elements to determine the structural capacity (empirical research). He investigated the structural capacity of columns. He concluded that the structural capacity is proportional to the cross section of the column (which is exactly right), and inversely proportional to the length of the column (which is almost correct). Ultimately, The German mathematician Leonard Euler (1707 – 1783) mathematically derived the formula for the buckling load. He found that a centrically loaded column with hinged ends has a buckling load of: 𝜋 2 𝐸𝐼 𝐹𝑘 = [𝑘𝑁] ℓ2 In which 𝐹𝑘 buckling load [kN] 𝐸 modulus of elasticity of the material [kN/m²] 𝐼 moment of inertia [m4] ℓ Length of the column [m] In general: 𝜋 2 𝐸𝐼 𝐹𝑘 = [𝑘𝑁] ℓ𝑘 2 In which ℓ𝑘 Buckling length of the column [m], Depending on the boundary conditions (Figure 2.18) 37 Chapter 2 - Columns 2.18 Four columns, equal in length, but with different boundary conditions. Buckling length ℓ𝑘 is the distance between bending points for a kinked rod. ℓ𝑘 = ℓ ℓ𝑘 = 0,7ℓ ℓ𝑘 = 0,5ℓ ℓ𝑘 = 2ℓ 𝐹𝑘 = 𝐹 𝐹𝑘 = 2𝐹 𝐹𝑘 = 4𝐹 𝐹𝑘 = 0,25𝐹 From the formula of Euler, we can derive the following: - The buckling load is proportional to the modulus of elasticity (thus depending on the material); Steel E = 210.000 N/mm² Concrete E = 30.000 N/mm² (Depending on concrete type, creep etc.) Timber E = 11.000 N/mm² (Depending on type of timber) - The buckling load is proportional to the moment of inertia (thus the shape of the cross section); - The buckling load is inversely proportional to the buckling length squared; - The buckling load depends on the boundary conditions (because ℓ𝑘 depends on the boundary conditions); - The buckling load is independent of the compressive strength of the material. 38  Chapter 2 - Columns 4. Designing columns Influence of the cross section on the buckling load For slender columns, where the buckling phenomenon plays a role, the shape of the cross section is vital. Figure 2.19 shows for a variety of sections (all with nearly the same cross sectional area A), the buckling load Fk, based on a centrically loaded column with hinged supports. Note: A high buckling load is favourable! 2.19 Influence of cross sectional shape on the buckling force. A = cross sectional area Iz = moment of inertia around the weak axis. Fk doorsnede vorm Area A 3771 3845 4029 3899 3864 3910 3880 3872 3888 3900 4000 [mm²] Iz 1464 631 1297 736 382 284 616 62 42 21 13 x 104 [mm4] From this graph can be conc