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CIV4445/6445 Lecture Notes 1

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CIV4445/6445 Lecture Notes 1. EARTHQUAKES and FUNDAMENTALS CONCEPTS of SEISMIC DESIGN 1.1. Introduction An earthquake can be defined as the ground shaking observed at the surface of the earth which is caused by the sudden release of energy in the Earth’s crust. Earthquakes can be caused by different processes, such as tectonic activities, volcanic eruptions, mining activities, the collapse of underground cavities and nuclear or chemical explosions. Tectonic earthquakes are particularly of interest to the structural engineers. Hazards associated with earthquakes are called as seismic hazards. The most important seismic hazards can be listed as ground shaking, structural hazards, liquefaction, and landslides. 1.2. Internal Structure of the Earth The crust is the outermost layer of the earth. The thickness of the crust varies from about 25 km to 40 km (it could be as thick as 60 to 70 km under some young mountain ranges) beneath the continents and to as thin as 5 km beneath the oceans. The crust is cooler than the materials below it. The oceanic crust is generally more uniform and denser than the continental crust. The boundary between the crust and the mantle, which is marked with a distinct change in wave propagation velocity, is called Moho discontinuity. The mantle, which lies below the crust, is about 2850 km thick. It is divided into two parts: upper mantle (~650 km thick) and lower mantle. The temperature is around 1000°C at the top of the upper mantle, around 3500°C at the base of the mantle. The mantle materials are in a viscous semi-molten state. They behave like a solid under rapidly applied stresses (such as seismic waves), whereas they flow slowly like a fluid when subjected to long-terms stresses. The earth’s outer core, which has a thickness of 2260 km, cannot transmit S-waves, as it is in a liquid state. The Gutenberg discontinuity marks the boundary between the mantle and the core, where S-wave velocity drops to zero. The inner core, which is in a solid state, is very dense with a specific gravity up to 15. Internal structure of the earth is shown in Figure 1.1. 1 CIV4445/6445 Lecture Notes Figure 1.1. Internal structure of the earth (Goe41, 2021) 1.3. Plate Tectonics The occurrence of earthquakes can be explained by the theory of plate tectonics. According to the Plate Tectonics, the earth’s crust is divided into six large, intact continental-sized blocks, aka plates (African, American, Antarctic, Australia-Indian, Eurasian, and Pacific), which move with respect to each other. There are also small plates. The relative deformation between plates occurs predominantly only in narrow zones near their boundaries either slowly and continuously, or suddenly in the form of earthquakes. Huge mass of the plates requires large driving forces, which can be explained with thermomechanical equilibrium of the earth’s materials as shown in Figure 1.2. A temperature gradient exists in the mantle due to the fact that the upper part of the mantle is contact with the relatively cold crust while lower part is in contact with the hot outer core. Therefore, cooler, denser materials rest on top of warmer, less dense materials. Eventually, cooler, denser material begins to sink under the action of the gravity, while the warm, less dense materials begins to rise. The sinking materials become warm and less dense and move laterally and begin to rise again as subsequently cooled material begins to sink. Therefore, horizontal components of these convection currents in the semi-molten rock of the mantle impose shear stresses on the bottom of the crust, dragging plates in various directions across the surface of the earth. This movement causes plates move apart from each other in some locations, and converges in others. Earthquakes are results of active tectonic movements predominantly concentrating near the plate boundaries (Figure 1.3). 2 CIV4445/6445 Lecture Notes Figure 1.2. Convection currents in the mantle (Brainly, 2021) Figure 1.3. Worldwide earthquakes of magnitude 5.5 and greater occurring between 1975 and 1999. (Encyclopædia Britannica, 2021) The characteristics of plate boundaries influence the nature of earthquakes occurring along them. There are three types of plate boundaries: a- Divergent Plate Boundaries (Spreading rift and ridge zones): In certain areas, plates move away from each other at the boundaries. These areas are called spreading ridge and rift zones (Figure 1.4). In ridges, molten materials from the mantle rises through the spreading boundaries to the surface of the crust and becomes part of the spreading 3 CIV4445/6445 Lecture Notes plates after cooling. Spreading normally ranges between 2-18 cm/year. Spreading ridges generally occur under oceans where sea-floor spreading creates new oceanic crust. Volcanic activity is very common in the vicinity of spreading ridge zones. The Pacific Ocean ridges (highest rate of spreading) and the mid-Atlantic ridges (lowest rate of spreading) can be given as examples for divergent plate boundaries. Sometimes, spreading occurs along the continental plates. The East African rift valley and Iceland can be given as examples for this kind of boundaries, where no crust is created. If rifting continues long enough, the continental rift turns into a normal mid-oceanic ridge. a) b) Figure 1.4. Divergent plate boundaries a) mid-Atlantic ridge and b) the East African rift valley (Newcomer, 2021) b- Convergent Plate Boundaries (Subduction zones): In convergent plate boundaries, adjacent tectonic plates move towards each other and collide. If one or both of the colliding plates is oceanic plate, a subduction zone will form. In these zones, the denser plate subducts beneath the other one. The plate above the subducting plate is called overriding plate. Convergent plate boundaries are often found near the edge of continents. 4 CIV4445/6445 Lecture Notes The Andes Mountain Range of western South America have formed by the subduction of the Nazca Plate (oceanic plate) beneath the South American plate (continental plate) as shown in Figure 1.5a. Two plates carrying continents moving towards each other do not subduct as their densities are not high and they form mountains. The Himalayan mountains and Tibetan plateau have formed as a result of the collision between the Indian-Australian Plate and Eurasian Plate as shown in Figure 1.5b. a) b) Figure 1.5. Divergent plate boundaries a) Andes mountains b) Himalayan mountains (Newcomer, 2021) An oceanic plate may collide with an oceanic plate. When the rate of plate convergence is high, a trench is formed at the junction where the two plates meet. Earthquakes occur 5 CIV4445/6445 Lecture Notes at the interface between the subducting and overriding plates (Benioff zones). The subducting plate melts due to high temperature inside the earth and magma is produced. Magma rising from the deeper part of a subduction zone may form volcanos that run parallel to these zones. c- Transform Plate Boundaries: The transform boundaries occur along the boundaries of plates that slide past each other without destroying old crust or creating new crust. Transform boundaries can offset mid-ocean ridges, subduction zones or both. The San Andreas Fault in California can be given as an example for a transform boundary that connects two spreading ridges. The Pacific Plate (on the west) moves north-westward relative to the North American Plate (on the east). Figure 1.6. Transform plate boundary a) Andes mountains b) Himalayan mountains (Newcomer, 2021) Figure 1.7 shows the tectonic plates and their boundaries worldwide. Inter-plate earthquakes (earthquakes occurring at the plate boundaries) correspond to 95% of worldwide seismic energy release. But, intra-plates earthquakes (earthquakes occurring inside the plates) cannot be explained with the plate tectonics. Shallow earthquakes occur at focal depths of 0-70 km, while intermediate events have focal depth of 70-300 km and deep earthquakes have focal depth of 300–700 km. Earthquakes occurring along the divergent plate boundaries are observed along the narrow zones close to the mid?oceanic ridges at moderate magnitudes and at a depth of less than 30 km. Along convergent plate margins with subduction zones, earthquakes range from shallow to depths of up to 700 km. Bands of earthquakes are wider along subduction zones, which can extend for more than 1,000 km because they take place 6 CIV4445/6445 Lecture Notes throughout the subducting slab that extends beneath the opposing plate. In fact, all of the very large earthquakes with magnitude 9 or higher take place at subduction boundaries because there is the potential for a greater width of rupture zone on a gently dipping boundary than on a steep transform boundary. Shallow earthquakes (usually less than 30 km deep) with large magnitudes (in the order of magnitude 8) can occur along transform faults in narrow bands close to plate margins. Where continents collide, earthquakes are scattered over a much wider area compared to earthquakes along mid-ocean ridges, transform margins, or subduction zones. Continental convergence earthquakes can be very damaging. Figure 1.7. Tectonic Plates with divergent, convergent and transform plate boundaries (Newcomer, 2021) 1.4. Elastic Rebound Theory When plates move with respect to each other, shear stresses increase along the plate boundaries with this movement and elastic strain energy is stored in the materials. If shear stress reaches the shear strength of the crustal material, accumulated energy is released suddenly and the strained material snap back towards equilibrium with the formation of a 7 CIV4445/6445 Lecture Notes rupture on a new or pre-existing and location along the boundary. The resulting rupture on the earth’s crust is named as a fault. If the material is weak and ductile, very limited amount of strain energy could be stored in the rock and released relatively slowly and aseismic movement is observed. If the rock is brittle and strong, the failure occurs rapidly by releasing energy in the form of heat and stress waves. This process is called as Elastic Rebound. These waves reach to the surface of the earth and are felt as earthquakes. 1.5. Fault Movement The fault mechanism can be defined with strike, dip and rake angles as shown in Figure 1.8. The intersection of a horizontal plane and a fault plane is called as strike of a fault. The angle between the strike vector and northerly direction is called azimuth of the strike ?. The angle between the horizontal plane and fault plane is called dip angle ?. The displacement of a point on the fault is called relative displacement ? ? . The angle between the direction of relative displacement and the horizontal direction is called rake ?. Figure 1.8. Geometric description of a fault plane Faults are classified according to their movement as dip-slip, strike-slip, and oblique faults. Fundamental fault movements are shown in Figure 1.9. In dip-slip faults, the fault movement mainly occurs in the vertical direction (in the direction of the dip). In a strikeslip fault, the fault movement takes place along the strike. In the oblique faults, the fault movement is a combination of dip-slip and strike-slip components (in the diagonal direction). The block underlying the fault plane is called foot-wall and the block above the fault plane is called hanging-wall. If the hanging wall in a dip-slip fault displaces downward direction relative to the foot wall, this fault is called normal fault. Normal faults create tensional forces in the crust. If the hanging wall in a dip-slip fault displaces upward direction relative to the foot wall, this fault is called reverse fault. Reverse faults create 8 CIV4445/6445 Lecture Notes compressional forces in the crust. If the dip angle of a reverse fault is so small, this fault is called as trust fault. Strike-slip faults, where fault plane has a nearly vertical orientation, can produce large movements and they are generally observed in transform boundaries. When an observer is located on one side of the fault, if the other side of the fault is moving to the left, this fault is called left lateral strike-slip fault. Similarly, an observer stands near to the right lateral strike slip fault, he/she would observe that the other side of the fault is moving to the right. Figure 1.9. Fundamental fault movements (Elnashai and Di Sarno, 2015) 9 CIV4445/6445 Lecture Notes The point on the fault plane where the rupture originates and seismic waves initiate is called the focus or hypocentre of an earthquake (Figure 1.10). The projection of the focus on the fault surface is called epicentre. The distance between a site and the epicentre is called epicentral distance and the distance between the focus and a site is called hypocentral distance. Figure 1.10. Description of earthquake location (Elnashai and Di Sarno, 2015) 1.6. Seismic Waves Fault ruptures cause brittle fractures of the earth’s crust and dissipate up to 10% of the total plate?tectonic energy in the form of seismic waves. When an earthquake occurs, different type of elastic seismic waves travel through the earth and they reflected and reflected at boundaries between different layers. The generated seismic waves follow different paths to reach different points on the earth’s surface. They are grouped into two categories: body waves and surface waves. Body waves can travel the interior of the earth. These are two types of body waves: P-waves and S-waves. The ground shaking felt at a site is generally a combination of these waves, especially at small distances from the source or ‘near?field’. P-waves (aka primary, compressional or longitudinal waves) generates successive compression and expansion of the materials that they travel through. A P-wave causes movements of the material particles to travel parallel to the travel direction. Deformation produced by P-wave is shown in Figure 1.11. P-waves can travel through solids and 10 CIV4445/6445 Lecture Notes fluids. They show small amplitudes and short periods and can be transmitted in the atmosphere. Speed of the body waves depends on the stiffness of the material they travel through. As the stiffness of the geologic materials is highest in compression, P-waves travel faster than other seismic waves, and they arrive first at a site, but P?waves have relatively little damage potential. Figure 1.11. Deformation produced by P-wave S-waves (aka secondary, shear or transverse waves) generates shearing deformation of the materials that they travel through. The particle movement in a material is perpendicular to the direction of the S-wave travel. Therefore, the motion of the S-wave can be separated into two components: SH-waves (horizontal particle movement) and SV-waves (vertical particle movement). An S-wave is slower than a P-wave and can only move through solid rock, not through any liquid medium. They show large amplitudes and long periods and both SH and SV-waves have significant damage potential on structures. Figure 1.12. Deformation produced by S-wave Surface waves are generated by the interaction of body waves with the surface and surficial layers of the earth. They propagate though the earth’s surface and their 11 CIV4445/6445 Lecture Notes amplitudes decrease exponentially with depth. These waves are more effective at locations farther from the source of the earthquake, as they are generated as a result of interaction. At distances greater than the twice of the thickness of the earth’s crust, surface waves will produce peak ground motions rather than body waves. For Earthquake Engineering, the most important surface waves are Rayleigh waves and Love waves. The interaction of P- and SV-waves with the earth surface produces the Rayleigh waves, which involve both vertical and horizontal particle motion. The interaction of SH-waves with soft surficial layer at the earth surface generates Lowe waves, which involve only horizontal particle motion. Figure 1.13. Deformation produced by Rayleigh wave Figure 1.14. Deformation produced by Lowe wave 1.7. Source to Site Effects The three important factors that affect ground motion characteristics are source, path and site effects (Figure 1.15). When a fault rupture occurs below the surface of the earth, rupture process affects the ground motion. After the rupture initiated, body waves travel away from the source in all directions until it reaches to the seismic bedrock at the site of interest. The characteristics of the seismic waves are altered by the geological units as 12 CIV4445/6445 Lecture Notes they travel from the source. This is called path effects. The site conditions near to the surface of the earth can significantly affect the amplitude, frequency content and duration of earthquake ground motions. This is known as site effect. Figure 1.15. Source, path and site effects (Kramer ) 1.8. Location of Earthquakes The location of an earthquake epicentre is obtained using the relative arrival times of pand s-waves. Minimum three seismographs are used in this process. Firstly, the difference in the arrival time of P- and S-waves ???−? is computed for the first seismograph as shown in Figure 16. Then, the distance between the hypocentre and the observation point is calculated using the following equation: ?= ???−? 1 1 ?? − ?? (1.1) where ?? and ?? are P- and S-wave velocities, respectively. In bedrock, P-wave velocities are generally in the range of 3-8 km/sec, while S-wave velocities vary between 2 to 5 km/sec. Figure 1.16. A typical seismogram (Bolt, 1978) 13 CIV4445/6445 Lecture Notes A circle of radius equal to the fist epicentral distance ? is plotted. Then, this procedure is repeated for the second and third seismographs. The intersection of these three circles show the approximate location of the earthquake (Figure 1.17). The accuracy of the procedure could be improved by increasing the number of seismographs. Figure 1.17. Initial prediction of earthquake location using wave arrival times from three stations (USGS, 2021) 1.9. Size of Earthquakes Before the development of the modern instrumentations, qualitative descriptions of the effects of earthquakes on human beings, animals and structures were used to characterise the size of the earthquakes. The invention of modern seismographs has enabled the development of quantitative measurement of earthquake size. Earthquake Intensity: Intensity is the oldest measure of the severity of the ground shaking. It provides a qualitative (non-instrumental) description of the effects of earthquakes in terms of human perceptions and reactions and damage level observed on the structures. Intensity is based on discrete scales. Several intensity scales, such as the Modified Mercalli Intensity (MMI) Scale, Mendvedev-Spoonheuer-Karnik scale (MSK 64), Japanese Meteorological Agency (JMA) intensity scale and European Macroseismic Scale (EMS), have been developed over the last a couple of centuries. European Macroseismic Scale (EMS) given in Table 1.1 is one of the most commonly used intensity scales. 14 CIV4445/6445 Lecture Notes Intensity maps can be developed by plotting the contour lines of equal intensity or ‘isoseismals’. These maps show the spatial variation of ground shaking and distribution of structural damage. Earthquake Magnitude: Magnitude is used to qualitatively determine the size of earthquakes based on some ground motion characteristic measured with seismic instruments. The Richter local magnitude, surface wave magnitude, body wave magnitude and moment magnitude are some of the most common magnitude scales. The Richter local magnitude uses trace amplitude measured with a specific seismometer. Amplitudes of Rayleigh waves and P-waves are used to define the surface wave and body wave magnitudes, respectively. The magnitude of large earthquakes may not be correctly determined using these magnitude scales, because the ground motion characteristics (amplitudes) may not change at the same rate as the size of the earthquake increases. Moment magnitude is based on the seismic moment, which is a measure of the work done by the earthquake and correlates well with the energy released during an earthquake. The seismic moment is defined as ? ?0 = ? ? ? (1.2) ? is the average where ? is the fault material rupture strength, ? is the rupture area and ? slip along the fault. The moment magnitude is obtained from ?? = log ?0 − 10.7 1.5 15 (1.3) CIV4445/6445 Lecture Notes Table 1.1. European Macroseismic Scale (EMS) (Grunthal, 1998) 1.10. Seismic Hazard Analysis and Design Ground Motion The level of ground shaking at a particular site or a region used in Earthquake-resistant Design Codes is usually determined with the aid of Probabilistic Seismic Hazard Analysis. Seismic Hazard Analyses involve the quantitative estimation of ground motion characteristics. Seismic Hazard Analyses require the identification and characterization of all potential sources of seismic activity that could produce significant ground motions at the site of interest. Earthquake sources may be identified on the basis of geologic, tectonic, historical, and instrumental evidence. In hazard analysis, the level of shaking 16 CIV4445/6445 Lecture Notes produced by an earthquake of a given size occurring at a given source-to-site distance is determined from ground motion prediction equations. In Earthquake-resistant Design Codes, elastic acceleration response spectra are often used to represent seismic loading for the seismic analysis of structures. As a result, design ground motions are often expressed in terms of elastic design spectra. Elastic design spectra provided in the seismic design codes and the response spectra of an actual earthquake are not the same. Response spectra from earthquakes are highly irregular and their shapes reflect the details of their specific frequency contents and phasing. Elastic design spectra, on the other hand, are generally quite smooth; they are usually determined by smoothing, averaging, or enveloping the response spectra of multiple motions. Figure 1.18 shows the comparison of Eurocode 8 Elastic Response spectra with response spectra of various recorded motions. The use of smooth elastic design response spectra implicitly recognizes the uncertainty with which soil and structural properties are known by avoiding sharp fluctuations in spectral accelerations with small changes in structural period. The response spectrum describes the maximum response of a single-degree-of-freedom (SDOF) system to a particular input motion as a function of the natural frequency (or natural period) and damping ratio of the SDOF system (Figure 1.19). Figure 1.18. Comparison of Eurocode 8 Elastic Response spectra with response spectra of various recorded motions (Nikolaou and Gilsanz, 2015) 17 CIV4445/6445 Lecture Notes Figure 1.19. Development of Elastic earthquake response spectra (Li, 2015) 1.11. Fundamentals Concepts of Seismic Design Earthquake Engineering is an engineering discipline which aims at mitigating the seismic risks by correctly predicting the earthquake effects imposed on structures and by designing, constructing and maintaining structures that have sufficient capacity to resist against these seismic effects without failure. The seismic design has substantially different principles than that of adopted for other types of actions. For quasi-static loads (such as dead, live loads, wind and snow loads etc.), the design forces are considered to be external (pre-determined). On the other hand, the seismic forces used for the design are not a result of the seismic input, but of the structural response to the seismic input. The same earthquake will generate different seismic forces in different structures. The structures who perform better in an earthquake are not necessarily stronger; the reason for their good behaviour could be the 18 CIV4445/6445 Lecture Notes relationship between their characteristics and the characteristics of the earthquake. For example, tall and slender buildings, with low natural frequencies, perform well in high frequency earthquakes. In other words, for a given earthquake excitation, the designer can control the seismic forces that develop in the structure by altering the characteristics of the structure. Earthquake excitations however are usually not “given”; they are generally unpredictable. So changing the stiffness is not always a good solution: a structure designed to perform well in a high-frequency earthquake, may be subjected to extreme loading in a low-frequency earthquake. Also, the structural response is mostly elastic under quasi-static loads until the maximum loads are reached. On the other hand, in seismic design, it is accepted that structures experience inelastic deformations and damage after certain level of seismic actions. 1.11.1. Structural Response Parameters Fundamental response parameters of structures controlling their earthquake response are stiffness, strength and ductility. Stiffness is a measure of resistance of a structural component or a group of structural components to deformation when subjected to a loading. Stiffness can be expressed as the ratio of stress (or force) resultants to strain (or displacement) resultants at a desired level of either of the two quantities. Stiffness is important parameter to satisfy serviceability requirements of structures under small earthquakes which frequently occurs. Low stiffness causes high deformability which substantially reduces the serviceably (functionality) of structures. Figure 1.20 shows a structure under horizontal loads together with a graph of total base shear vs top displacement response of the structure. ?? is the stiffness at deformation ?? and force resistance ?? (?? = ?? /?? ). We need to consider lateral and vertical stiffness of structures. Structures that are designed considering gravity loads are in general have sufficient vertical stiffness. Earthquakes generate lateral inertia forces on the structures which are dominant on horizontal forces, therefore horizontal stiffness is primary importance. Strength is the ability of a structural component or a group of structural components to resist applied loads without failure. Strength controls the level of inelastic deformations of structures when subjected to mid-size earthquakes. In Figure 1.20, ?? and ?? are the force capacities corresponding to ?? and ?? , respectively. ?? is the yield strength which corresponds to the yield displacement ?? . 19 CIV4445/6445 Lecture Notes Ductility is the ability of a structural component or a group of structural components to deform beyond yield limit without substantially reducing its bearing capacity. Ductility ? is expressed as the ratio of a maximum value of a deformation quantity to the value of the same quantity at yield level. Collapse prevention under very rare earthquakes is controlled by ductility. For the structure shown in Figure 1.20, the ductility is expressed by ?= a) ?? ?? (1.4) b) Figure 1.20. a) A structure under horizontal loads b) total base shear vs top displacement response of the structure (Courtesy of Elnashai, 2015) Demand is the force or deformation imposed on a component or a group of components due to earthquake ground motions. Capacity refers to the strength, stiffness and ductility available in a component or a group of components to resist earthquake ground motions. In other words, capacity represents the response of the structure to the demand. Earthquake demand on structures is not constant and varies with the change in the ground motion and structural characteristics during inelastic response. Capacity may also vary with the change in the structural and ground motions characteristics during inelastic response. In the inelastic response regime, demand and capacity are coupled. Damping is another important quantity for the structural response and is consequential to the three fundamental response parameters of structures (stiffness, strength and ductility). Structural damping is always beneficial for structures when subjected to seismic 20 CIV4445/6445 Lecture Notes loads. Increased damping results in reduced dynamic amplification of the input motion; and the reduced accelerations in the structure result in reduced design forces (force demand) in the elements. For example, if the structure is designed to withstand the seismic forces without any damage (i.e. to remain linear-elastic), the designer will select large cross sections and a lot of reinforcement, so that the element resistances are larger than the element forces obtained in the linear-elastic analysis. Since no damage occurs in the structure, the seismic energy that enters the structure will be transformed into potential (elastic) energy which will then be transferred to the ground. This is a strengthbased approach, which was the basis for the old seismic design codes (up to the 1980s). If the structural response is linear elastic, the only loss of energy is due to strains in the materials which results in low damping (about 5% of critical in RC frames), a relatively large floor accelerations and large seismic (or inertial) forces acting on each floor. However, if the designer allows some damage, for example cracking of concrete and yielding of reinforcement in the beams, then some of the energy that enters the structure will be spent on damaging the structure, which produces hysteretic damping that significantly increases the effective structural damping (could be over 10%). The floor accelerations and the resulting seismic forces acting on the structure will be reduced. It is important to note that while some damage of the structural elements is allowed, they are not allowed to fail. The elements should be designed with limited load capacity but increased capacity to deform while carrying the design load, i.e. the elements are designed to behave in ductile manner. 1.11.2. Performance Levels The philosophy of the seismic design of structures is based on the performance levels that appropriately engineered structures should satisfy for different seismic intensity levels. These performance level can be summarized as follows: - prevent near collapse, serious damage or loss of lives in rare major earthquake events (Ultimate Limit State seismic action or ULS seismic action) - prevent structural damage and minimize non-structural damage in less frequent moderate earthquake; - prevent damage of non-structural components in very frequent earthquakes. 21 CIV4445/6445 Lecture Notes 1.11.3. Capacity Design Method In seismic design of structures, it is generally not economical or possible to design all elements of a structure to behave in a ductile manner. Therefore, a dissipative (ductile) structure consist of both dissipative (ductile) elements with high inelastic deformation capacity and non-dissipative (brittle) elements with high load-carrying capacity. Certain critical regions of the dissipative elements, which are called plastic hinges, are properly designed and detailed to maintain high energy dissipation of the structure under severe deformation conditions. The strength of these critical regions is arranged by prioritising the yielding of such regions. The remaining structural elements in the structure are designed in a way to prevent failure by providing them with strength greater than that corresponding development of maximum feasible strength (based on overstrength) in the potential plastic hinge regions. This design method is called the Capacity Design Method. The Capacity Design Method prevents the formation of storey mechanisms shown in Figure 1.21 (structure on the right). Storey mechanisms are formed when plastic hinges occur in all columns of one floor. The moments in the column are equal and opposite to the sum of the moments that develop in the adjacent beams. If the column is designed so that its ultimate moment (resistance) is larger than the sum of the resistance moments of the adjacent beams, then plastic hinges will develop in the beams and the storey mechanism will be avoided (Figure 1.21 structure on the left). This design strategy is also known as plastic hinge hierarchy or simply strong column-weak beam. 22 CIV4445/6445 Lecture Notes Figure 1.21. Ductile and brittle behaviour of a structure 1.11.4. Ductile/Brittle Behaviour and Energy Dissipation Ductility is a general characteristic of inelastic behaviour and it can be described in terms of material (concrete, steel, wood, rubber), simple structural elements (beams, columns, walls), groups of elements (beam-column connection, coupled wall) or complex structures (frame, braced frame, wall-frame system). The ductility of a material depends on the material characteristics and loads. For example, unreinforced concrete in tension is brittle, whereas steel is ductile. However, a steel bar in compression is brittle, because of buckling. In structural elements ductility depends on materials, loading and detailing. For example, unreinforced concrete beam in bending is brittle, whereas reinforced concrete beam is ductile, because of yielding of the steel reinforcement intension. But, if the section is over-reinforced, then the concrete in the compressed zone will fail before the reinforcement yields in tension. The result will be a brittle failure. The ductility of structural elements is local ductility. Even if all structural elements (beams/columns) in a frame building are ductile, the structure as a whole will behave in a brittle manner if all plastic 23 CIV4445/6445 Lecture Notes hinges occur simultaneously in one floor, resulting in formation of a storey mechanism (Figure 1.21 structure on the right) and a brittle failure of the structure. The ductile response (or all elements and the structure as a whole) results in energy dissipation: part of the energy that enters the system is spent on yielding of steel, cracking of concrete and other damage. The elasto-plastic force-displacement (or moment-rotation or stress-strain) relationship under cyclic loading is called hysteretic curve; the area enclosed by the curve in each load-unload cycle is hysteretic energy, and the resulting damping in the structure is hysteretic damping. To achieve large energy dissipation, a structural element (or the whole structure), has to possess sufficient strength and ductility; the energy is a product of force and displacement. 1.11.5. Dissipative Structures Energy dissipation is a very important issue in the seismic behaviour of structures. The reason for this is that the earthquake loading is cyclic. If the seismic input is a harmonic with a frequency equal to one of the natural frequencies of the structure, then the response of the structure is in resonance with the excitation. If the structure possesses no damping, then its response will increase with each cycle until the stresses in the elements exceed their resistance (no matter how strong they are designed to be). In reality, even when the structural response is considered linear elastic, the structures posses some damping (viscous damping) due to straining and friction within the material or friction in the connections between different elements of the structure. For steel and reinforced concrete frames the coefficient of viscous damping for the first mode of vibration is typically 3-5%. If the response of the structure is not elastic, i.e. when plastic deformations develop in some structural members, then part of the work that caused these deformations is not converted into potential (elastic) energy but dissipated (spent) in the structure. This energy is called hysteretic energy (Figure 1.22a) and the associated damping hysteretic damping. The damping ratios which result from the hysteretic energy dissipation are typically much larger than the viscous damping and can reach 20% of critical. The influence of hysteretic damping on seismic response can be illustrated by the simulation of response of a dissipative and non-dissipative 6-floor concrete frame subjected to harmonic excitation (Figure 1.22b). 24 CIV4445/6445 Lecture Notes a) b) Figure 1.22. a) Linear elastic and nonlinear behaviour with hysteretic energy dissipation, b) influence of hysteretic damping on response of structures. 1.11.6. Eurocode 8 Seismic Design Guidelines Eurocode 8 provides guidance for ductility-based design, by prescribing: 1. Analyses and seismic design actions for calculating the design value (?? , where ? could be moments, axial or shear force); the elements are then designed using the relevant codes (EC2, EC3..) to provide sufficient resistance (?? ≤ ?? , where ?? is the design resistance); 2. Capacity design rules, or rules for providing a hierarchy of resistances in the structure so that the damage is distributed throughout the structure (and not concentrated in one part) and in selected elements (e.g beams, not columns) so that the overall behaviour of the structure is ductile (not brittle); 3. Detailing rules for local ductility of elements (beams, columns, walls ..), based on experimental research. 25 CIV4445/6445 Lecture Notes References: Brainly, 2021, Access Date: March 4, 2021, Elnashai A. S. and Di Sarno L., 2015, Fundamentals of Earthquake Engineering: From Source to Fragility, John Wiley and Sons. Encyclopædia Britannica, Access Date: March 4, 2021, Geo41, 2021, Access Date: March 4, 2021, Grünthal G., 1998, European Macroseismic Scale 1998. Cahiers du Centre Europèen de Gèodynamique et de Seismologie. Conseil de l’Europe, Conseil de l’Europe Kramer S. L., 1996, Geotechnical Earthquake Engineering, Prentice Hall. Li, B., 2015, Response Spectra for Seismic Analysis and Design, https://www.semanticscholar.org/paper/Response-Spectra-for-Seismic-Analysis-andDesign-Li/beadeae9fce9ef7aaa84c885d21f13a293f4d4ec#paper-header Newcomer, 2021, Access Date: March 4, 2021, 50 > 250 180 - 360 15-50 70 - 250 < 180 < 15 < 70 formation, including at most 5 m of weaker material at the surface. ? Deposits of very dense sand, gravel, or very stiff clay, at least several tens of metres in thickness, characterised by a gradual increase of mechanical properties with depth. ? Deep deposits of dense or medium dense sand, gravel or stiff clay with thickness from several tens to many hundreds of metres. ? Deposits of loose-to-medium cohesionless soil (with or without some soft cohesive layers), or of predominantly soft-to-firm cohesive soil. ? A soil profile consisting of a surface alluvium layer with vs values of type C or D and thickness varying between about 5 m and 20 m, underlain by stiffer material with vs > 800 m/s. 38 CIV4445/6445 Lecture Notes Table 3.1. Ground types (cont.) Ground Description of stratigraphic Type profile ?1 Parameters Deposits consisting, or containing a ??,30 ???? ?? (m/s) (blows/30cm) (kPa) 40) and high water content ?2 Deposits of liquefiable soils, of sensitive clays, or any other soil profile not included in types A – E or ?1 For sites with ground conditions matching either one of the two special ground types ?1 or ?2 , special studies for the definition of the seismic action are required. For these types, and particularly for ?2 , the possibility of soil failure under the seismic action shall be taken into account. Note: Special attention should be paid if the deposit is of ground type ?1. Such soils typically have very low values of ?? , low internal damping and an abnormally extended range of linear behaviour and can therefore produce anomalous seismic site amplification and soil-structure interaction effects (see Eurocode 8-5 (2004) Section 6). In this case, a special study to define the seismic action should be carried out, in order to establish the dependence of the response spectrum on the thickness and ?? value of the soft clay/silt layer and on the stiffness contrast between this layer and the underlying materials. 3.2. Seismic Action 3.2.1. Seismic Zones 39 CIV4445/6445 Lecture Notes The seismic zones are defined by the National Authorities depending on the local hazard. By definition, the hazard within each zone is assumed to be constant. In Eurocode 8, the hazard is described in terms of a single parameter, i.e. the value of the reference peak ground acceleration on type ? ground (rock), ??? . Additional parameters may require for specific types of structures and these are given in the relevant Parts of Eurocode 8. Note: The reference peak ground acceleration on type ? ground, ??? , for use in a country or parts of the country, may be derived from zonation maps found in its National Annex. The reference peak ground acceleration, chosen by the National Authorities for each seismic zone, corresponds to the reference return period ?NCR of the seismic action for the no-collapse requirement (or equivalently the reference probability of exceedance in 50 years, ?NCR ). An importance factor ?I equal to 1,0 is assigned to this reference return period. For return periods other than the reference, the design ground acceleration on type A ground ?? is equal to ??? times the importance factor ?I as explained in Section 3.5.3 (?? = ?I ??? ). For regions of low seismicity (ag < 0.08 g, or those where the product ag S < 0.1 g), reduced or simplified seismic design procedures for certain types or categories of structures may be used. For regions of very low seismicity (ag < 0.04 g, or those where the product ag S < 0.05 g), the provisions of Eurocode 8 need not be observed. 3.2.2. Basic Representation of Seismic Action In Eurocode 8, the earthquake motion at a given point on the surface is represented by an elastic ground acceleration response spectrum (Elastic Response Spectrum). The shape of the elastic response spectrum is taken as being the same for the two levels of seismic action for the no-collapse requirement (ultimate limit state – design seismic action) and for the damage limitation requirement. The horizontal seismic action is described by two orthogonal components assumed as being independent and represented by the same response spectrum. For the three components of the seismic action, one or more alternative shapes of response spectra may be adopted, depending on the seismic sources and the earthquake magnitudes generated from them. 40 CIV4445/6445 Lecture Notes When the earthquakes affecting a site are generated by widely differing sources, the possibility of using more than one shape of spectra should be considered to enable the design seismic action to be adequately represented. In such circumstances, different values of ?? will normally be required for each type of spectrum and earthquake. For important structures (?I >1,0) topographic amplification effects should be taken into account. Time-history representations of the earthquake motion may be used (see Section 3.2.3). Allowance for the variation of ground motion in space as well as time may be required for specific types of structures (such as bridges, silos, tanks, pipelines, towers and chimneys). 3.2.2.1. Horizontal Elastic Response Spectrum The shape of the elastic response spectrum is represented in Figure 3.1. For the horizontal components of the seismic action, the elastic response spectrum ?? (?) is defined by the following expressions: ? ?? (?) = ?? ? [1 + ? (2.5 ? − 1)] ? ?? (?) = ?? ? 2.5 ? for 0 ≤ ? ≤ ?? for ?? ≤ ? ≤ ?? (3.1) ?? ?? (?) = ?? ? 2.5 ? [ ? ] ?? (?) = ?? ? 2.5 ? [ ?? ?? ?2 ] for ?? ≤ ? ≤ ?? for ?? ≤ ? ≤ 4 ? where ?? (?) is ordinate of the elastic response spectrum, ? is the vibration period of a linear single-degree-of-freedom system, ?? is the design ground acceleration on type A ground (?? = ?I ??? ), ?? is the lower limit of the period of the constant spectral acceleration branch; ?? is the upper limit of the period of the constant spectral acceleration branch; ?? is the value defining the beginning of the constant displacement response range of the spectrum; 41 CIV4445/6445 Lecture Notes ? is the soil factor; ? is the damping correction factor with a reference value of ? = 1 for 5% viscous damping. The values of the periods ?? , ?? and ?? and of the soil factor ? describing the shape of the elastic response spectrum depend upon the ground type. Figure 3.1. Shape of the elastic response spectrum Figure 3.2 shows “constant spectral acceleration”, “constant spectral velocity” and “constant displacement” regions of the elastic spectrum. 42 CIV4445/6445 Lecture Notes Figure 3.2. “Constant spectral acceleration”, “constant spectral velocity” and “constant displacement” regions of the elastic spectrum If deep geology is not accounted for, Type 1 and Type 2 spectra can be used. If the earthquakes that contribute most to the seismic hazard defined for the site for the purpose of probabilistic hazard assessment have a surface-wave magnitude, ?? , not greater than 5,5, it is recommended that the Type 2 spectrum is adopted. For the five ground types A, B, C, D and E the recommended values of the parameters ?? , ?? , ?? and ? are given in Table 3.2 for the Type 1 Spectrum and in Table 3.3 for the Type 2 Spectrum. Table 3.2. Values of the parameters describing the recommended Type 1 elastic response spectra ? ?? (s) ?? (s) ?? (s) A 1.0 0.15 0.4 2.0 B 1.2 0.15 0.5 2.0 C 1.15 0.20 0.6 2.0 D 1.35 0.20 0.8 2.0 E 1.4 0.15 0.5 2.0 Ground type Table 3.3. Values of the parameters describing the recommended Type 2 elastic response spectra ? ?? (s) ?? (s) ?? (s) A 1.0 0.05 0.25 1.2 B 1.35 0.05 0.25 1.2 C 1.5 0.10 0.25 1.2 Ground type 43 CIV4445/6445 Lecture Notes D 1.8 0.10 0.30 1.2 E 1.6 0.05 0.25 1.2 Figure 3.3 and Figure 3.4 show the shapes of the recommended Type 1 and Type 2 spectra, respectively, normalised by ?? , for 5% damping. Figure 3.3. Recommended Type 1 elastic response spectra for ground types A to E (5% damping) 44 CIV4445/6445 Lecture Notes Figure 3.4. Recommended Type 2 elastic response spectra for ground types A to E (5% damping) For ground types ?1 and ?2 , special studies should provide the corresponding values of ?? , ?? , ?? and ?. The elastic displacement response spectrum, ?De (?), is obtained by direct transformation of the elastic acceleration response spectrum, ?? (?), using the following expression: ? 2 ??? (?) = ?? (?) [ ] 2? (3.2) Equation (3.2) should normally be applied for vibration periods not exceeding 4,0 s. 3.2.2.2. Vertical Elastic Response Spectrum The vertical component of the seismic action is represented by an elastic response spectrum, ?ve (?), derived using following expressions ? ??? (?) = ??? [1 + ? (3.0 ? − 1)] for ? 45 0 ≤ ? ≤ ?? (3.3) CIV4445/6445 Lecture Notes ??? (?) = ??? 3.0 ? ? ??? (?) = ??? 3.0 ? [ ??] ??? (?) = ??? 3.0 ? [ ?? ?? ?2 ] for ?? ≤ ? ≤ ?? for ?? ≤ ? ≤ ?? for ?? ≤ ? ≤ 4 ? Similar to the spectra defining the horizontal components of the seismic action,If the earthquakes that contribute most to the seismic hazard defined for the site for the purpose of probabilistic hazard assessment have a surface-wave magnitude, ?? , not greater than 5,5, it is recommended that the Type 2 spectrum is adopted. For the five ground types ?, ?, ?, ?, and ? the recommended values of the parameters ?? , ?? , ?? and ? describing the vertical spectra are given in Table 3.4. The parameters of the vertical response spectrum are independent of the ground type, but applicable only for ground types ?, ?, ?, ?, and ?. In case of special ground types (?1 and ?2 ), site investigations are required. Table 3.4. Recommended values of parameters describing the vertical elastic response spectra ??? /?? ?? (s) ?? (s) ?? (s) Type 1 0.90 0.05 0.15 1.0 Type 2 0.45 0.05 0.15 1.0 Spectrum 3.2.2.3. Design Ground Displacement The design ground displacement ?? corresponding to the design ground acceleration, may be estimated by means of the following expression ?? = 0.025 ?? ? ?? ?? 3.2.2.4. Design Spectrum for Elastic Analysis The capacity of structural systems to resist seismic actions in the non-linear range generally permits their design for resistance to seismic forces smaller than those corresponding to a linear elastic response. To avoid explicit inelastic structural analysis 46 CIV4445/6445 Lecture Notes in design, the capacity of the structure to dissipate energy, through mainly ductile behaviour of its elements and/or other mechanisms, is taken into account by performing an elastic analysis based on a response spectrum reduced with respect to the elastic one, henceforth called a ''Design Spectrum''. This reduction is accomplished by introducing the behaviour factor ?. The behaviour factor ? is an approximation of the ratio of the seismic forces that the structure would experience if its response was completely elastic with 5% viscous damping, to the seismic forces that may be used in the design, with a conventional elastic analysis model, still ensuring a satisfactory response of the structure. The values of the behaviour factor ?, which also account for the influence of the viscous damping being different from 5%, are given for various materials and structural systems according to the relevant ductility classes in the various Parts of Eurocode 8. The value of the behaviour factor ? may be different in different horizontal directions of the structure, although the ductility is the same in all directions. For the horizontal components of the seismic action the design spectrum, ?? (?), shall be defined by the following expressions: 2 ? 2.5 2 ?? (?) = ?? ? [3 + ? ( ? − 3)] for ? ?? (?) = ?? ? ?? (?) { ?? (?) { 2.5 2.5 ?? [ ] ? ? ?? ≤ ? ≤ ?? for ?? ≤ ? ≤ ?? (3.4) ≥ ? ?? = ?? ? for ? = ?? ? 2.5 ? [ ?? ?? ?2 0 ≤ ? ≤ ?? ] ?? ≤ ? ≤ 4 ? for ≥ ? ?? where ? is the lower bound factor for the horizontal design spectrum. The recommended value for ? is 0,2. Other parameters are defined in Section 3.2.2.1. For the vertical component of the seismic action the design spectrum is given by Equation (3.4) with the design ground acceleration in the vertical direction, ??? replacing ?? , ? taken as being equal to 1,0 and the other parameters as defined in Section 3.2.2.2. 2 ? 2.5 2 ??? (?) = ??? [3 + ? ( ? − 3)] ? for 47 0 ≤ ? ≤ ?? (3.5) CIV4445/6445 Lecture Notes ??? (?) = ??? ??? (?) { ??? (?) { 2.5 ? = ??? 2.5 ? ?? ≤ ? ≤ ?? for ?? ≤ ? ≤ ?? 2.5 ?? ? [?] ≥ ? ??? = ??? for [ ?? ?? ?2 ] ≥ ? ??? ?? ≤ ? ≤ 4 ? for For the vertical component of the seismic action a behaviour factor ? up to 1.5 should generally be adopted for all materials and structural systems. The adoption of values for ? greater than 1.5 in the vertical direction should be justified through an appropriate analysis. 3.2.3. Alternative Representations of the Seismic Action 3.2.3.1. Time-history Representation The seismic motion may also be represented in terms of ground acceleration timehistories and related quantities (velocity and displacement). When a spatial model of the structure is required, the seismic motion consist of three simultaneously acting accelerograms. The same accelerogram may not be used simultaneously along both horizontal directions. Simplifications are possible in accordance with the relevant parts of Eurocode 8. Depending on the nature of the application and on the information actually available, the description of the seismic motion may be made by using artificial accelerograms and recorded or simulated accelerograms. Artificial accelerograms shall be generated so as to match the horizontal and vertical elastic response spectra defined in Sections 3.2.2.1 and 3.2.2.2, respectively, for 5% viscous damping (? = 5%). The duration of the accelerograms should be consistent with the magnitude and the other relevant features of the seismic event underlying the establishment of ?? . When site-specific data are not available, the minimum duration ?? of the stationary part of the accelerograms should be equal to 10 ?. The number of accelerograms used in the analysis should be sufficient to provide a stable statistical measure (mean and variance) of the response quantities. The amplitude and 48 CIV4445/6445 Lecture Notes the frequency content of the time histories must provide a level of reliability equal to the one achieved by using the Elastic Response Spectra. The suite of artificial accelerograms should observe the following rules: a) a minimum of 3 accelerograms should be used; b) the mean of the zero period spectral response acceleration values (calculated from the individual time histories) should not be smaller than the value of ?? ? for the site in question. c) in the range of periods between 0.2 ?1 and 2 ?1, where ?1 is the fundamental period of the structure in the direction where the accelerogram will be applied; no value of the mean 5% damping elastic spectrum, calculated from all time histories, should be less than 90% of the corresponding value of the 5% damping elastic response spectrum. Recorded accelerograms, or accelerograms generated through a physical simulation of source and travel path mechanisms, may be used, provided that the samples used are adequately qualified with regard to the seismogenetic features of the sources and to the soil conditions appropriate to the site, and their values are scaled to the value of ?? ? for the zone under consideration. The suite of recorded or simulated accelerograms to be used should also observe above listed rules. 3.2.3.2. Spatial Model for Seismic Actions For special structures (e.g. long bridges, dams) where the excitation may vary at different support points a spatial model is required for the seismic action. The models should be established on the basis of the wave propagation principles and consistent with the Elastic Response Spectra. 3.2.4. Combinations of the Seismic Action with Other Actions The design value ?? of the effects of actions in the seismic design situation shall be determined in accordance with EN 1990:2002, Section 6.4.3.4. The masses should be calculated by taking into account the presence of the masses associated with all gravity loads appearing in the following combination of actions: ∑ ??,? " + " ∑ ??,? ? ??,? 49 (3.6) CIV4445/6445 Lecture Notes where ??,? are permanent loads, ??,? are variable loads and ??,? is the combination coefficient for variable action ? (see 4.2.4, ??,? = ? ?2,? ). The combination coefficients ??,? take into account the likelihood of the loads ??,? not being present over the entire structure during the earthquake. These coefficients may also account for a reduced participation of masses in the motion of the structure due to the non-rigid connection between them. Values of ??,? are given in EN 1990:2002. 50 CIV4445/6445 Lecture Notes References Eurocode 8-1 (EN 1998?1), 2004, Design of structures for earthquake resistance – Part 1: General rules, Seismic Actions and Rules for Buildings, European Committee for Standardization (CEN): Brussels, Belgium. Eurocode 8-5 (EN 1998?5), Design of structures for earthquake resistance – Part 5: Foundations, Retaining Structures and Geotechnical Aspects, European Committee for Standardization (CEN): Brussels, Belgium. 51 CIV4445/6445 Lecture Notes 4. DESIGN OF BUILDINGS 4.1. General The scope of Section 4 is the general rules for earthquake resistant design of buildings. The rules should be used in conjunction with the specific rules for buildings from different materials, given in Sections 5-9: (5) concrete, (6) steel, (7) steel-concrete composite buildings, (8) timber and (9) masonry. Guidance for base isolated buildings are given in Section 10. 4.2. Characteristics of Earthquake Resistant Buildings Post?earthquake field observation has shown that buildings with irregular configurations are more vulnerable than their regular ones. 4.2.1. Basic Principles of Conceptual Design In seismic regions, the aspect of seismic hazard should be taken into account in the early stages of the conceptual design of a building, thus enabling the achievement of a structural system which, within acceptable costs, satisfies the fundamental requirements (no-collapse and damage limitation requirements). The guiding principles governing this conceptual design are: - Structural simplicity: Structural simplicity, which is characterised by the existence of clear and direct paths for the transmission of the seismic forces, is an important objective to be pursued, since the modelling, analysis, dimensioning, detailing and construction of simple structures are subject to much less uncertainty and thus the prediction of its seismic behaviour is much more reliable. Note: Compact, convex and closed shapes perform better than complex, concave and open sections. In addition, dimensioning, detailing and construction of simple structures are often more cost?effective than for complex structural systems. - Uniformity, symmetry and redundancy: Uniformity in plan is characterised by an even distribution of the structural elements which allows short and direct transmission of the inertia forces created in the distributed masses of the building. If necessary, uniformity may be realised by subdividing the entire building by seismic joints into dynamically independent units, provided that these joints are designed against pounding of the individual units. 52 CIV4445/6445 Lecture Notes Uniformity in the development of the structure along the height of the building is also important, since it tends to eliminate the occurrence of sensitive zones where concentrations of stress or large ductility demands might prematurely cause collapse. A close relationship between the distribution of masses and the distribution of resistance and stiffness eliminates large eccentricities between mass and stiffness. If the building configuration is symmetrical or quasi-symmetrical, a symmetrical layout of structural elements, which should be well-distributed in-plan, is appropriate for the achievement of uniformity. The use of evenly distributed structural elements increases redundancy and allows a more favourable redistribution of action effects and widespread energy dissipation across the entire structure. - Bi-directional resistance and stiffness: Horizontal seismic motion is a bi-directional phenomenon and thus the building structure must be able to resist horizontal actions in any direction. To satisfy this, the structural elements should be arranged in an orthogonal in-plan structural pattern, ensuring similar resistance and stiffness characteristics in both main directions. The choice of the stiffness characteristics of the structure, while attempting to minimise the effects of the seismic action (taking into account its specific features at the site) should also limit the development of excessive displacements that might lead to either instabilities due to second order effects or excessive damages. - Torsional resistance and stiffness: Besides lateral resistance and stiffness, building structures should possess adequate torsional resistance and stiffness in order to limit the development of torsional motions which tend to stress the different structural elements in a non-uniform way. In this respect, arrangements in which the main elements resisting the seismic action are distributed close to the periphery of the building present clear advantages. - Diaphragmatic behaviour at storey level: In buildings, floors (including the roof) play a very important role in the overall seismic behaviour of the structure. They act as horizontal diaphragms that collect and transmit the inertia forces to the vertical structural systems and ensure that those systems act together in resisting the horizontal seismic action. The action of floors as diaphragms is especially relevant in cases of complex and 53 CIV4445/6445 Lecture Notes non-uniform layouts of the vertical structural systems, or where systems with different horizontal deformability characteristics are used together (e.g. in dual or mixed systems). Floor systems and the roof should be provided with in-plane stiffness and resistance and with effective connection to the vertical structural systems. Particular care should be taken in cases of non-compact or very elongated in-plan shapes and in cases of large floor openings, especially if the latter are located in the vicinity of the main vertical structural elements, thus hindering such effective connection between the vertical and horizontal structure. Diaphragms should have sufficient in-plane stiffness for the distribution of horizontal inertia forces to the vertical structural systems in accordance with the assumptions of the analysis (e.g. rigidity of the diaphragm), particularly when there are significant changes in stiffness or offsets of vertical elements above and below the diaphragm. - Adequate foundation: With regard to the seismic action, the design and construction of the foundations and of the connection to the superstructure must ensure that the whole building is subjected to a uniform seismic excitation. For structures composed of a discrete number of structural walls, likely to differ in width and stiffness, a rigid, box-type or cellular foundation, containing a foundation slab and a cover slab should generally be chosen. For buildings with individual foundation elements (footings or piles), the use of a foundation slab or tie-beams between these elements in both main directions is recommended. 4.2.2. Primary and Secondary Seismic Members A certain number of structural members (e.g. beams and/or columns) may be designated as “secondary” seismic members (or elements), not forming part of the seismic action resisting system of the building. The strength and stiffness of these elements against seismic actions must be neglected. They do not need to conform to the requirements of Sections 5 to 9. Nonetheless these members and their connections must be designed and detailed to maintain support of gravity loading when subjected to the displacements caused by the most unfavourable seismic design condition. Due allowance of 2nd order effects (P-? effects) should be made in the design of these members. Sections 5 to 9 give rules, in addition to those of EN 1992, EN 1993, EN 1994, EN 1995 and EN 1996, for the design and detailing of secondary seismic elements. All structural members not designated as being secondary seismic members are taken as being 54 CIV4445/6445 Lecture Notes primary seismic members. They are taken as being part of the lateral force resisting system, should be modelled in the structural analysis in accordance with 4.3.1 and designed and detailed for earthquake resistance in accordance with the rules of Sections 5 to 9. The total contribution to lateral stiffness of all secondary seismic members should not exceed 15% of that of all primary seismic members. The designation of some structural elements as secondary seismic members is not allowed to change the classification of the structure from non-regular to regular as described in 4.2.3. 4.2.3. Criteria for Structural Regularity The structures must be categorised as regular and non-regular for the design purpose. Note: In building structures consisting of more than one dynamically independent units, the categorisation and the relevant criteria refer to the individual dynamically independent units. This distinction has the following implications on the design: - the structural model can be either a simplified planar or spatial model; - the method of analysis can be either a simplified response spectrum analysis (lateral force procedure, based on first mode approximation) or a multi-modal; - the value of the behaviour factor ?, which should be decreased for buildings non-regular in elevation if either (1) the geometric nonregularity or (2) the non-regular distribution of over-strengths exceeds the prescribed limits. For non-regular in elevation buildings the decreased values of the behaviour factor ? are given by the reference values multiplied by 0,8. The consequences of the structural regularity are summarised in Table 4.1. The reference behaviour factor ? values are given in Sections 5-9. Specific conditions given in Section 4.3.3.2 should be met to use the lateral force method. 55 CIV4445/6445 Lecture Notes Table 4.1. Consequences of structural regularity on seismic analysis and design Regularity a Allowed Simplifications Behaviour factor ? Plan Elevation Model Linear-elatic analysis (for linear analysis) Yes Yes Planar Lateral Force Reference value Yes No Planar Multi-modal Reduced value No Yes Spatiala Lateral Force Reference value No No Spatial Multi-modal Reduced value Under the specific conditions given in Section 4.3.3 a separate planar model may be used in each horizontal direction. Criteria for Regularity in Plan: A structure is considered regular in plan, if all the conditions listed below are satisfied: ? The distribution of lateral stiffness and mass is approximately symmetrical in plan with respect to two orthogonal axes (see Figure 4.1a for examples of symmetrical and non-symmetrical lateral stiffness). ? The plan configuration is compact. This means that each floor is delimited by a polygonal convex line. If there are set-backs in plan, regularity in plan may still be considered as being satisfied, provided that these setbacks do not affect the floor inplan stiffness; and that area of setbacks ?? must be less than 5% of the floor area ?? (see Figure 4.1b). ? The in-plan stiffness of the floors is large enough in comparison with the lateral stiffness of the vertical structural elements, so that the floor displacements are equally distributed among the vertical elements. In this respect, the L, C, H, I, and X plan shapes should be carefully examined, notably as concerns the stiffness of the lateral branches, which should be comparable to that of the central part, in order to satisfy the rigid diaphragm condition (see Figure 4.1c for an example of insufficient in-plan stiffness: external columns deform more than those in the centre). ? The slenderness ? = ???? ???? of the building in plan is not higher than 4, where ???? and ???? are respectively the larger and smaller in plan dimension of the building, measured in orthogonal directions (see Figure 4.1d). 56 CIV4445/6445 Lecture Notes ? At each level and for each direction of analysis ? and ?, the structural eccentricity ??? and the torsional radius ?? should satisfy the following conditions: ??? ≤ 0.30 ?? and (4.1) ?? ≥ ?? where ??? is the distance between the centres of mass and lateral stiffness measured in the direction normal to the direction of the analysis (see Figure 4.1e), and ?? = √ ?? ⁄? ? where ?? is global torsional stiffness and ?? is global lateral stiffness in the ?-direction, ?? is the radius of gyration of the floor mass in plan (see Figure 4.1f). ?? = √ ∑ ?? ?? 2 ⁄∑ ? (4.2) ? ∑ ?? ?? 2 stands for the polar moment of inertia of the floor mass in plan with respect to the centre of mass of the floor and ?? is the mass radius of gyration with respect to the centre of mass of the floor. In single storey buildings the centre of stiffness is defined as the centre of the lateral stiffness of all primary seismic members. For the classification of structural regularity in plan and for the approximate analysis of torsional effects, in multi-storey buildings, approximate definitions of the centre of stiffness and of the torsional radius are possible using the moments of inertia of the cross sections of the vertical elements, if the following two conditions are satisfied: a) all lateral load resisting systems, such as cores, structural walls, or frames, run without interruption from the foundations to the top of the building; b) the deflected shapes of the individual systems under horizontal loads are similar. This condition may be considered satisfied in the case of frame systems and wall systems. In general, this condition is not satisfied in dual systems (see Figure 4.1g). In frames and in systems of slender walls with prevailing flexural deformations, the position of the centres of stiffness and the torsional radius of all storeys may be calculated as those of the moments of inertia of the cross-sections of the vertical elements. If, in addition to flexural deformations, shear deformations are also significant, they may be accounted for by using an equivalent moment of inertia of the cross-section. 57 CIV4445/6445 Lecture Notes Figure 4.1. Regularity in Plan Criteria for Regularity in Elevation: A structure is considered regular in plan, if all the conditions listed below are satisfied: ? All lateral resisting systems such as cores, structural walls, or frames, run without interruption from the foundation to the top of the building. ? Both the lateral mass and stiffness of the individual storeys remain constant or decrease gradually from the base to the top, without abrupt changes. 58 CIV4445/6445 Lecture Notes ? In frame buildings the ratio of the actual storey resistance to the resistance required by the analysis does not vary significantly between adjacent storeys. ? If there are elevation setbacks, their size is within the below given limits: a) For gradual setbacks preserving axial symmetry, the setback at any floor should not be greater than 20 % of the previous plan dimension in the direction of the setback (see Figure 4.2a and Figure 4.2b); b) For a single setback within the lower 15 % of the total height of the main structural system, the setback should not be greater than 50 % of the previous plan dimension (see Figure 4.2c). In this case the structure of the base zone within the vertically projected perimeter of the upper storeys should be designed to resist at least 75% of the horizontal shear forces that would develop in that zone in a similar building without the base enlargement; c) If the setbacks do not preserve symmetry, in each face the sum of the setbacks at all storeys should not be greater than 30 % of the plan dimension at the ground floor above the foundation or above the top of a rigid basement, and the individual setbacks should not be greater than 10 % of the previous plan dimension (see Figure 4.2d). 4.2.4. Combination Coefficients for Variable Actions The combination coefficients for variable action should be computed as ??,? = ? ?2,? where: • ?2,? is the combination coefficients for the quasi-permanent value of variable action ?. These are given in EN 1990:2002. Example: for imposed load in buildings ?2 = 0.3 for residential and office areas, ?2 = 0.6 for shopping and congregation areas, ?2 = 0.8 for storage areas. For snow loads ?2 = 0.2 for Sweden and Finland, ?2 = 0 for Greece and Spain. For wind load ?2 = 0. • ? is a coefficient which depends on the type of variable action, occupancy and position of the storey, as shown in Table 4.2. Example: office building floor (category B): ? = 0.8, ?2 = 0.3, if the imposed load is ??,? = 2.5 ??/?2 ; since ?2 = 0 for wind, and assuming ?2 = 0 for snow, the variable load that will be combined with the permanent load during earthquake is ∑ ?? ??,? = 0.8 ? 0.3 ? 2.5 ??/?2 . 59 CIV4445/6445 Lecture Notes Figure 4.2. Criteria for regularity of buildings with setbacks Table 4.2. Coefficient ? for calculating ??,? Type of variable action (as defined in EN 1991-1-1) Categories: (A) domestic and residential (B) offices, (C) areas where people congregate (schools, restaurants, cinemas, museums) Storey ? Roof 1.0 Storeys with correlated occupancy 0.8 Storeys with independent occupancy 0.5 60 CIV4445/6445 Lecture Notes Categories D-F (shopping, industrial, parking) and archives 1.0 4.2.5. Importance classes and importance factors See section 2.5.4. 4.3. Structural analysis 4.3.1. Modelling The structural model should adequately represent the mass and stiffness distribution of the building, so that all the significant deformation shapes and inertia forces are considered in the analyses. Note: This is reflected in the allowed simplifications in the analysis of structures with different irregularities (see Table 4.1). It means that if, for example, the structure is irregular in plan, then a planar model (2D) will not give an adequate representation of its mass and stiffness distribution, and a spatial (3D) model has to be used. If non-linear analysis is used, in addition to mass and stiffness, the model must also represent the strength distribution in the structure. The model should also account for the contribution of joint regions to the deformability of the building, e.g. the end zones in beams or columns of frame type structures. Nonstructural elements, which may influence the response of the primary seismic structure, should also be accounted for. In general, the structure may be considered to consist of (i) vertical load resisting system (ii) lateral load resisting systems, connected by (iii) horizontal diaphragms. The in-plane stiffness of the floors shall be large enough in comparison with the lateral stiffness of the vertical structural elements, so that a rigid diaphragm behaviour may be assumed.If the floors are sufficiently rigid in their planes, the masses and the moments of inertia of each floor may be lumped at the centre of gravity, thus reducing the dynamic DOFs to three per floor. Note: The diaphragm is considered as rigid, if, when it is modelled with its actual in-plane flexibility, its horizontal displacements nowhere exceed those resulting from the rigid diaphragm assumption by more than 10% of the corresponding absolute horizontal displacements in the seismic design situation. 61 CIV4445/6445 Lecture Notes The stiffness of concrete buildings, steel-concrete composite buildings and masonry buildings should be calculated for a state of cracks in tensile regions (at the initiation of yielding of the reinforcement). Unless a more accurate analysis of the cracked elements is performed, the elastic flexural and shear stiffness properties of concrete and masonry elements should be taken as 50% of the uncracked stiffness. Infill walls which contribute significantly to the lateral stiffness and resistance of the building should be taken into account. The deformability of the foundation should be taken into account in the model, whenever it may have an adverse overall influence on the structural response. Note: Foundation deformability (including the soil-structure interaction) may always be taken into account, including the cases in which it has beneficial effects. The masses should be calculated from the gravity loads appearing in the combination of actions indicated in Section 3.2.4. The combination coefficients ??,? are given in 4.2.4. 4.3.2. Accidental Torsional Effects In order to account for uncertainties in the mass locations, the calculated centre of gravity for each floor ? should be considered as being displaced from its nominal position in a spatial model in each direction by an accidental eccentricity ??? : ??? = ±0.05 ?? (4.3) where ?? is the floor dimension perpendicular to the direction of the seismic action. 4.3.3. Methods of Analysis The seismic effects and the effects of the other actions included in the seismic design situation can be determined by the means of a linear-elastic analysis. The reference method for determining the seismic effects is the multi-modal response spectrum analysis, using a linear-elastic model of the structure and the design spectrum given in Section 3.2.2.4. Depending on the structural characteristics of the building one of the following two types of linear-elastic analysis may be used: (a) the “lateral force method of analysis” for buildings meeting the conditions given in Section 4.3.3.2; and (b) the “multi-modal response spectrum analysis", which is applicable to all types of buildings. 62 CIV4445/6445 Lecture Notes As an alternative to a linear method, a non-linear method may also be used, such as (c) non-l...