An Introduction to Earthquake Engineering (BEG 454CI) B.E civil IV/I Acme Engineering college (Purbanchal University)

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1 An introduction to Earthquake engineering An Introduction to Earthquake Engineering (BEG 454CI) B.E civil IV/I Acme Engineering college (Purbanchal University) Er. Adarsha Thapa Acme Engineering College 1 Downloaded from

2 An introduction to Earthquake engineering Chapter one Introduction 1.1 Effects of earthquakes (on buildings) An earthquake event can create different hazards for a structure. The effect that is of primary interest to a designer is the inertial response of structures to ground acceleration that occurs during an earthquake. Ground shaking during an earthquake generates inertial forces in the structure. This inertial force is in response to the self weight of the structure undergoing acceleration from an initially rest position. (Newton's second law of motion). Earthquake induced ground motions create inertial forces significantly in the lateral direction by shaking the structures back and forth. Horizontal and vertical shaking Earthquake causes shaking of the ground in all three directions - along the two horizontal directions (N-S, E-W) and the vertical direction (up-down). Structures are primarily designed to carry gravity loads(vertical loads), i.e. they are designed for a force equal to the mass (this includes mass due to self weight and imposed loads) times the acceleration due to gravity in the vertical downward direction. The vertical acceleration during ground shaking either adds to or subtracts from the acceleration due to gravity. Since factors of safety are used in the design of structures to resist the gravity loads, usually most structures tend to be adequate against vertical shaking. o However, horizontal shaking along the N-S, E-W directions is usually of major concern to the designer. Structures designed for gravity loads, in general, may not be able to safely sustain the effects of horizontal earthquake shaking. Hence it is often necessary to design a structure for resisting the significant lateral loads that can arise during an earthquake. 1.2 Theories and criteria of earthquake design General as well as specific theories (principles) and criteria for earthquake design are given in IS 1893(part I) 'Criteria for Earthquake Resistant Design of structures'. Other additional criteria are also given by design codes such as; IS4326- 'Earthquake resistant design and construction of buildings', IS 'Ductile detailing of reinforced concrete structures' etc. Some of the important principles(theory) and criteria given in IS1893(section 6) are; I. The design philosophy adopted in the code is to ensure that structures possess at least a minimum strength to; i. resist minor earthquake (<DBE*) which may occur frequently, without damage; 2 Downloaded from

3 An introduction to Earthquake engineering ii. resist moderate earthquake (DBE) without significant structural damage, though some non-structural damage may occur iii. resist major earthquake (MCE*) without collapse *(Design Basis Earthquake (DBE) is defined as the maximum earthquake that reasonably can be expected to be experienced at the site once during the lifetime of the structure. The earthquake corresponding to the ultimate safety requirements if often called as Maximum Considered Earthquake (MCE). DBE and MCE are classified based on probability.) II. Actual forces that appear on structures during earthquakes are much higher than the design forces specified in the code. It is recognized that the complete protection against earthquakes of all sizes is not economically feasible and design based alone on strength criteria is not justified. The basic criteria of earthquake resistant design should be based on lateral strength as well as deformability and ductility capacity of structure with limited damage, but no collapse. Ductility in the structures will arise from inelastic material behaviour and detailing of reinforcement in such a manner that brittle failure is avoided and ductile behaviour is induced by allowing steel to yield in a controlled manner. Hence the gap between actual and design forces is reduced by utilizing the additional reserve strength available due to ductility of the structure. III. The design lateral forces specified in the code shall be considered in each of the two orthogonal directions of the structure. For structures which have lateral force resisting elements in the two orthogonal directions only, the design lateral force shall be considered along one direction at a time, and not in both directions simultaneously. 1.3 Basic requirements for earthquake resistant structures The important characteristics that any earthquake resistant structure should possess are; 1) Adequate stiffness and strength 2) Ductility and toughness 3) Regularity 4) Continuous load path 5) Redundancy 6) Stable foundations Adequate stiffness and strength: Strong earthquakes will induce both vertical and lateral forces in a structure. The lateral forces that that tend to move structures horizontally have proven to be particularly damaging. If a structure has inadequate lateral stiffness or strength, these lateral forces can produce large horizontal displacements in the structure and potentially cause instability. o Hence a structure should possess adequate strength to resist both gravity and lateral loads. o The structure should also possess adequate lateral stiffness to limit the deflections. Ductility and toughness: Ductility and toughness are structural properties that relate to the ability of a structural element to sustain damage when overloaded while continuing to carry 3 Downloaded from

4 An introduction to Earthquake engineering load without failure. These are extremely important properties for structures designed to sustain damage without collapse. o Most structural elements are designed to provide sufficient strength to support anticipated loads without failure and enough stiffness so that they will not deflect excessively under these loads. If such an element is subjected to a load substantially larger than it was designed to carry, it may fail in an abrupt manner, losing loadcarrying capacity and allowing the structure to collapse. o Masonry and concrete will crush when overloaded in compression and will crack and pull apart when placed in tension or shear. o Wood will crush when overloaded in compression, will split when overloaded in shear, and will break when overloaded in tension. o Steel will buckle if overloaded in compression and will twist when overloaded in bending if not properly braced, but will yield when overloaded in tension. When steel yields, it stretches a great deal while continuing to carry loads, and this property allows it to be used in structures of all types to provide them with ductility and toughness. Regularity: A structure is regular if the distribution of its mass, strength, and stiffness is such that it will sway in a uniform manner when subjected to ground shaking. Hence for a regular structure the lateral movement in each storey(elevation), and on each side of the structure (plan) will be about the same. o Regular structures tend to dissipate the earthquake's energy uniformly throughout the structure resulting in relatively light but well distributed damage. o In an irregular structure, however, the damage can be concentrated in one or a few locations, resulting in extreme local damage and a loss of the structure's ability to survive the shaking. Continuous load path: It is very important that all parts of a building or structure, including non-structural components, be tied together to provide a continuous path that will transfer the inertial forces resulting from ground shaking from the point of origination to the ground. If all the components of a building or structure are not tied together in this manner, the individual pieces will move independently and can pull apart, allowing partial or total collapse to occur. Redundancy: If all of a structure's strength and resistance is concentrated in only one or a few elements, the structure will not have any residual strength if these elements are seriously damaged and it could collapse. If a structure is redundant, a relatively large number of elements participate in providing a structure's strength and, if only a few are badly damaged, the remaining elements may have adequate residual strength to prevent collapse. Stable foundations: In addition to being able to support a structure's weight without excessive settlement, the foundation system must be able to resist earthquake induced overturning forces and be capable of transferring large lateral forces between the structure and the ground. On sites that can be subjected to liquefaction or lateral spreading, it is important to provide vertical bearing support for the foundation beneath the liquefiable layers of soil. 4 Downloaded from

5 An introduction to Earthquake engineering Chapter 2.0 Fundamental of Earthquake Engineering Causes of Earthquakes Ground motion generated by sudden displacement within the earth's crust is called an earthquake. Earthquakes may result from a number of natural and human-induced phenomenon. Natural causes include meteoric impact, and volcanic activity. Human related activities that can cause earthquakes are underground nuclear explosions, rock stress changes induced by the filling of large reservoirs, and excavation in mines. However, the vast majority of damaging earthquakes originate at, or adjacent to, the boundaries of crustal tectonic plates, due to relative deformations at the boundaries. Plate tectonics: According to the widely accepted theory of plate tectonics, the crust and upper part of the mantle of the earth, called the lithosphere, is about 50Km thick under the deepest oceans and 150Km thick under the highest mountains. The lithosphere is subdivided into tectonic plates, which moves as rigid bodies on a relatively soft asthenosphere due to the convection currents that circulate as a result of the temperature and pressure difference between the core and the crust. Tectonic plates Because of the relative displacements between the two adjoining plates, high stresses are induced in the bedrock materials within the affected zones. In the case where the stresses exceed the material strength, or the frictional capacity at the plate interface, the accumulated strain energy is released in the form of earthquake shock waves. 5 Downloaded from

6 An introduction to Earthquake engineering There are three different types of plate boundaries: 1. Divergent (constructive) 2. Convergent (destructive) 3. Transform(conservative) Mechanism of earthquakes (Elastic rebound theory) The Elastic rebound theory helps to explain the mechanism of an earthquake. According to this theory, the strain energy is accumulated in a rock material along a fault due to relative deformation of the adjacent rocks. After a long period of time, between two earthquakes, the strain energy reaches its ultimate limit after which the rock fractures and slips back towards a stress-free state. This slippage releases the tremendous amount of strain energy that had accumulated due to the distortion in the rock. Elastic rebound theory 6 Downloaded from

7 An introduction to Earthquake engineering Faults: A fault is defined as a fracture or crack in the rock along which some definite movement has taken place. Earthquakes are mainly caused due to sudden slip at faults which results in shaking of earth. This sudden slip results in release of large amount of energy. Based on fault movement, faults can be classified into four main categories, 1. Normal dip-slip fault 2. Reverse dip-slip fault 3. Strike-slip fault 4. Oblique slip fault Focus (Hypocenter) of an earthquake: The place of origin of earthquake in the interior of the earth is known as the focus or hypocenter of an earthquake. The focus is the rupture point within the earth's crust and it represents the source of emission of energy. Epicenter: The point on the earth's surface that lies vertically above the focus is called the Epicenter. Focal depth is the depth of the focus below the Epicenter. Focal distance is the distance from the focus to a given reference point. Epicentral distance is the distance from the Epicenter to a given reference point. 7 Downloaded from

8 An introduction to Earthquake engineering Seismic Waves: The large amount of energy released during an earthquake causes radial propagation of waves within the earth. These waves are called seismic waves which transmit energy from one point to another through different layers of soil and rock. The waves reflect and refract on their way to the earth's surface. Seismic waves can be classified into two main categories: 1. Body wave 2. Surface wave Body waves: Body waves are generated at the rupture zone and they propagate within the earth's interior. There are two types of body waves; i. Primary waves(p-waves) ii. Secondary Waves (S-waves) Primary waves: These are longitudinal waves which vibrate in the same direction as the propagation of the wave. They travel in a push-pull manner similar to sound waves. The main characteristics of P-waves are ; P-waves travel faster than S-waves and are thus first to arrive at a recording station They travel through all forms of matter (Solid, liquid, and gas) The velocity of P-waves depend on the density and compressibility of the medium Secondary waves (S-waves): These are transverse or shear waves which cause particles to vibrate in perpendicular direction to the propagation of the wave. They are slower than P- waves and hence arrive at a recording station after P-waves. Other important characteristics of S-waves are; 8 Downloaded from

9 An introduction to Earthquake engineering S-waves travel only through solids and not through fluids (liquid and gas) since fluids do not have any shear resistance. Their velocity depends on the density and shear strength of material through which they pass. Surface waves: Surface waves are the result of reflection and refraction of P and S-waves during propagation in stratified formation of the earth's crust. They travel along the surface of the earth and are of two types; i. Love waves ii. Rayleigh waves Love waves: They vibrate in a horizontal plane parallel to the earth's surface and perpendicular to the direction of wave propagation. Love waves travel faster than Rayleigh waves and cannot travel through fluids Love waves along with S-waves cause more damage to structures Rayleigh waves: These are surface waves that vibrate in an elliptical orbit in the vertical plane. Measure of earthquakes (quantification) The size of an earthquake is measured by determining its magnitude and intensity. Magnitude: The magnitude of an earthquake is determined on the basis of the amount of energy released at the earthquake focus. Hence, magnitude is a quantitative measure of an earthquake. One of the widely used magnitude scale is the Richter scale, proposed by Charles Richter in A single earthquake event can have only one magnitude an hence the magnitude of an earthquake is independent of the location at which the measurement is obtained. 9 Downloaded from

10 An introduction to Earthquake engineering Intensity: An earthquake can also be measured based on its effect on people, built environment, and the natural environment. This observed effect on the surface of the earth's crust during and after an earthquake event is known as intensity of an earthquake. Hence, Intensity is a qualitative measure of an earthquake. For the same magnitude earthquake, the intensity can vary widely at different locations. One of the most widely used intensity scale is the Modified Mercalli Intensity scale (MMI). Other Intensity scales include European Macroseismic scale(ems), Medvedev-Sponheuer-Karnik (MSK). Modified Mercalli Intensity scale (MMI scale) Intensity Acceleration(cm/s 2 ) Damage Potential I Less than 1 only instruments can record II Over 1 felt only on upper floors III Over 2.5 felt by people at rest and indoor IV Over 5 felt by people in motion, glass panes of window rattle. V Over 10 (0.01g) cracking of plaster, people are disturbed VI Over 30 people panic and run outdoors, perceptible damage to buildings VII Over 100 (0.1g) well-designed and well constructed structures are affected IX Over 300 well-designed and well constructed buildings are badly damaged X Over 650 destruction of large number of well-designed and wellconstructed buildings, damage to roads and rails, cracks in grounds, sliding of slopes, liquefactions XI Over 1000 (g) most of the buildings are destroyed, damage to bridges and dams XII Over 3000 total destruction, waves seen on ground surface, rivers courses altered, large amount of rocks may move Types of Earthquakes (classification) Earthquakes can be classified on the following basis; 1. Based on depth of focus: Shallow earthquakes - Depth < 70 Km Intermediate earthquakes - Depth between Km Deep earthquakes - Depth>300 Km 2. Based on Magnitude: Great if M>8.0 Major if M = Strong if M = Moderate if M= Light if M = Minor if M = Micro if M < Based on location: Inter-plate - earthquakes that occur along the tectonic plate boundaries 10 Downloaded from

11 An introduction to Earthquake engineering Intra-plate- earthquakes occurring away from plate boundaries 4. Based on epicentral distance (Δ): Local earthquake - Δ < 1 degree Regional - Δ < 1 to 10 degrees Teleseismic - Δ > 10 degrees Measuring magnitude of an earthquake using different available methods I. Richter magnitude (M L ) : The Richter magnitude is also known as local magnitude. The scale was defined by Charles Richter in According to Richter, the magnitude of an earthquake M is given by a logarithm of a maximum displacement amplitude A (in µm), recorded on a Wood-Anderson seismograph located at exactly 100Km from the Epicenter. The standard Wood-Anderson seismograph has a natural frequency of 1.25 Hz, a critical damping ratio of 0.8 and amplification factor of M L = log 10 A log 10 A 0 = log 10 ( A A 0 ) where, A is recorded earthquake displacement amplitude in µm for A0 is amplitude for zero magnitude earthquake (equal to 1 µm epicentral distance of 100Km) II. Body Wave Magnitude (m B ): The body wave magnitude is based on amplitude of P- waves and can be expressed as; m B = log 10 ( A ) + σ(, h) T max where, A is ground displacement amplitude in µm for P-wave T is time period of P-waves in seconds Δ is epicentral distance in degrees h is focal depth of the location of seismograph recording III. Surface wave magnitude (M s ): The Ms scale is used to quantify stronger earthquakes. It is based on the recordings of amplitudes of surface waves with a period T of 20±2 seconds. A commonly used equation for computing Ms of a shallow focus (<50km) earthquake from seismograph records between epicentral distances (20 < < 160 ) can be given as; M s = log 10 ( A s T ) max log Downloaded from

12 An introduction to Earthquake engineering where, As is the displacement amplitude of surface wave in µm T, is the period of surface waves in seconds Δ is the epicentral distance in degrees IV. Moment magnitude scale (M W ): The ML, mb,and MS scales are all based on using the maximum recorded amplitude of seismic waves to determine the magnitude of an earthquake event. These three scales have been found to under-estimate the energy released during a very large earthquake event. Hence, seismologists have developed a standard magnitude scale, known as the moment magnitude scale, which can be more accurate for large earthquakes. The moment magnitude of an earthquake is calculated using the amount of moment released during an earthquake. The seismic moment released depends on the physical dimension of the rupture (A), shear strength of the rock(µ), and the average displacement of the fault plane(d). Hence, the seismic moment is given by; M 0 = μ. A. d The moment magnitude, MW, can be determined using the following equation; M W = 2 3 [ log 10M 0 (dyne. cm) 16.0 ] (note: 1 Nm = 10 7 dyne.cm, or 1N=10 5 dyne) # An earthquake causes an average of 2.6m strike-slip displacement over a 75km long, 22km deep portion of a transform fault. Assuming the rock along the fault has an average rupture strength of 180KPa, estimate the seismic moment released and hence the moment magnitude of the earthquake. Solution: 12 Downloaded from

13 An introduction to Earthquake engineering Average fault displacement, d = 2.6 m Length of fault, L = 75 Km Depth of fault, D = 22 Km Average rupture strength, µ = 180 KPa = N/m 2 We have, Seismic moment released, M 0 = μ. A. d = ( ) ( ) 2.6 = Nm = ( ) 10 7 = dyne-cm Hence, Moment magnitude, M W = 2 [ log 3 10M ] = 2 [ log 3 10( ) 16.0 ] = 3.93 M W = 3.93 # A seismograph located 1200 Km from the Epicenter of an earthquake, records a maximum ground displacement of 15.6mm for surface waves having a period of 20 s. Based on these assumptions, determine the surface wave magnitude of the earthquake. Solution: We have, maximum ground displacement amplitude, As = 15.6mm = µm Time period of Surface waves, T = 20 s Total circumference of the earth = 40,076 Km Hence, Km Km = Epicentral distance in degrees, = Downloaded from

14 An introduction to Earthquake engineering Surface magnitude, M s = log 10 ( A s T ) max log M s = log 10 ( ) log (10.78 ) M S = 7.9 Time history (response history) The time history is the ground motion record that is, plot of the acceleration, velocity, and displacement, of a point on the ground surface as a function of time for the entire duration of earthquake. The maximum amplitude of the recorded acceleration is termed as the peak response acceleration. Similarly the peak response velocity, and the peak response displacement, are the maximum amplitude of the recorded velocity and displacement respectively Chi-Chi earthquake, Taiwan. recorded ground acceleration history (NS-component) 14 Downloaded from

15 An introduction to Earthquake engineering Response Spectra Displacement response history The response spectrum is a standard method of representation of the response of structures to ground acceleration. It is the summarisation of response histories in terms of the peak response of all SDOF systems with different natural periods. Hence, the response spectrum is a plot of peak value of a response quantity such as acceleration, velocity, or displacement of an SDOF system as a function of natural vibration period Tn of the system with a particular damping ratio. Several such plots for different values of damping ratios can be included to cover the range of damping values included in actual structures. Acceleration response spectra from 1985 Chile earthquake Design Response Spectrum The response spectrum for actual ground motion is quite irregular and hence the individual spectrum is not convenient for use in design. The design response spectrum is based on the statistical analysis of the response spectra for the ensemble of ground motions which is smooth and representative. The design response spectrum is not intended to match 15 Downloaded from

16 An introduction to Earthquake engineering the response spectrum of any particular ground motion, but it represents the average of response spectra of several ground motions. The design response spectrum is supposed to cover the wide range of natural periods and the practical range of damping values so that it provides the peak response of all possible structures. The design response spectrum is presented in the seismic standards. In the Indian standard, the acceleration response spectrum is presented for 5% damping with the modification factors for other values of damping. IS 1893 (2002) - Design Response Spectrum for 5% damping Earthquake force parameters The ground motion characteristics at a location during an earthquake is useful in assessing the seismic demand on structures built near the location. The strong motions are measured by accelerographs. The record or time history of ground acceleration gives the accelerogram. The accelerographs are generally influenced by the type of seismic action, focal distance, path of wave propagation, orientation of the site with respect to the fault line, local site conditions and topography. The peak values of ground acceleration, velocity, and displacement, are known as amplitude parameters. These amplitude parameters give an idea of the severity of shaking at a site. Peak Ground Acceleration The peak horizontal acceleration (PGA) is the most commonly used measure of strength of ground shaking at a site. PGA relates directly to the maximum inertial forces generated in the structure. PGA is normally expressed as a fraction of gravitational acceleration; for example, the PGA of 0.981m/s 2 is expressed as 0.1g (where g is acceleration due to gravity). 16 Downloaded from

17 An introduction to Earthquake engineering The horizontal component of ground acceleration, PHA, absolute value of the horizontal acceleration recorded at a site. is taken as the largest Horizontal component of acceleration is primarily used to report ground motion as structures are designed for vertical loads and margin of safety in the vertical direction are usually adequate for earthquake induced vertical load. Attenuation The reduction in vibration of earthquake waves as they travel further away from their Epicenter is called attenuation. Attenuation Laws: A mathematical description of the behaviour of characteristics of earthquake ground motion as a function of the distance from source of energy is known as attenuation law. Attenuation relationships are usually derived from analysis of small and moderate earthquakes. Several different attenuation laws have been proposed in literature. Amplitude parameters (PGA) have been used to express vibration loss as earthquake waves travel away from their source. The amplitude parameters for a site are given as varying according to the magnitude of the earthquake and the focal distance of the site. This can be stated as PGA = f(m,r), where M is the magnitude of earthquake in Richter scale and R is the focal distance. Esteva (1969) proposed the following attenuation law for earthquake effect at a site with firm ground, a = 5600.e0.8M (R+40) 2 where, a is PGA in cm/sec2 V = 32.eM (R+25) 2 where, V is PGV in cm/sec 2 d = ( V2 R0.6). a where, d is PGD in cm Cornell et.al (1979) proposed the following relationship, ln[pga(cm/sec 2 )] = M 1.8 ln (R + 25) # Find the maximum possible PGA at a site 29Km from an earthquake source that is capable of producing a maximum of 7.6 magnitude earthquake in Richter scale. Use the attenuation law proposed by Cornell et.al. solution: We have, focal distance, R = 29 km max magnitude of earthquake, M = Downloaded from

18 An introduction to Earthquake engineering Using Cornell et.al proposed attenuation law, ln[pga(cm/sec 2 )] = M 1.8 ln (R + 25) ln[pga] = ln ( ) ln[pga] = PGA = e = cm/sec 2 PGA = = 0.449g cm/sec2 Seismic risk and seismic zoning Seismicity is the geographic and historic distribution of earthquake as recorded from different Epicenter location on the earth's surface. A seismic zone is designated where areas with similar seismicity are classified in the same zone. Hence a seismic zone can be an area of similar seismicity sharing a common factor, e.g. the Himalayan belt that lies between the convergence of the Indian and Eurasian Plate. The goal of seismic zoning is to mark and separate regions of similar probable intensity of ground motion in a country. This helps in providing a guideline for earthquake resistance design in constructed facilities. Hence, a seismic zoning map provides the probable PGA in a region based on the history of earthquakes within an area. The map can also indicate the magnitude, intensity, and recurrence interval or frequency of earthquakes in the area where a structure is to be designed. Seismic hazard: It is the probability that an earthquake will occur in a given geographical area, within a given window of time, and with ground motion intensity exceeding a given threshold. Seismic Risk: It is estimated based on the seismic hazard in a given area and the risk factor calculated can be used in building codes, planning infrastructure projects, and determining insurance rates. In addition, seismic risk also depends on susceptibility of the structure to damage and the consequences of damage. Earthquake hazards: Earthquakes pose several hazards to our natural and built environment such as; 18 Downloaded from

19 An introduction to Earthquake engineering 1. Ground shaking - seismic waves cause damage to structures 2. Ground failure - soil liquefaction, landslides, ground settlement etc 3. Tsunami - long period sea waves produced by sea floor movement 4. Life-line hazards - destruction of water supply system, communication and electricity grid etc. Liquefaction: The phenomenon in which the soil liquefies and the strength of soil is drastically reduced to the point where the soil is unable to support the structure. Liquefaction mainly occurs in sandy soil with high water table. 19 Downloaded from

20 An introduction to Earthquake engineering Chapter3 Introduction to Structural dynamics Dynamic loading: A loading which changes in direction and/or magnitude with respect to time is a dynamic loading. Hence, earthquake loads, blast impacts, mechanical vibrations, all have to be considered as dynamic loads in order for their effects on structures to be understood. Response of structures to vibration: Structural responses are the effects due to the dynamic loading on the structure. The structural responses such as displacements and velocities for a structure under dynamic loading will also be dynamic, i.e. they will vary with time. Hence structural dynamics involves finding the responses of a structure to dynamic loading. Two different approaches can be used to carry out dynamic analysis. The choice of method used depends upon how the loading is defined. 1. Deterministic approach: The analysis of the response of any specified structural system to a prescribed dynamic loading is defined as a deterministic analysis. Prescribed dynamic loading: A loading for which the time variation is fully known, even though the load may be highly oscillatory or irregular in character, is known as a prescribed dynamic load. 2. Non-deterministic approach: The analysis of response of any specified structural system to a random dynamic loading is known as a nondeterministic loading. Random dynamic loading: A loading for which the time variation is not completely known but can be defined in a statistical sense is termed as a random dynamic loading. Types of prescribed dynamic loadings Prescribed loads can be classified into two categories ; 1. Periodic loading 2. Non-periodic loading 1. Periodic loading: A periodic loading has the same loading pattern or time variation for a large number of loading cycles. The simplest periodic loading is the sinusoidal variation, which is also known as simple harmonic loading. 2. Non-periodic loading: Non-periodic loads do not have a consistent repeating pattern as shown by a periodic load. Non-periodic loads may either be short duration impulsive loads, or longer duration general forms of loads. 20 Downloaded from

21 An introduction to Earthquake engineering Simple Harmonic motion: A motion for which the acceleration of the system is directly proportional to its displacement from a mean position and is directed to that mean position can be said to be in simple harmonic motion. e.g. a pendulum. Simple harmonic motion of a pendulum 21 Downloaded from

22 An introduction to Earthquake engineering Damping: The property of a structural system primarily due to its material and structural arrangement, by which free vibration steadily diminishes in amplitude is called damping. The simplest model of damping in a system is the viscous damping model. For viscous damping, the velocity of the system is directly proportional to the damping force. The proportionality constant is the damping coefficient 'C'. Hence, damping force, F D = C. V where, F D is the damping force acting on the system C is the damping coefficient V is the velocity of the system Degree of freedom: The number of independent co-ordinates required to fully describe the motion of a structural system is known as its degree of freedom. For instance, the motion of a single pendulum shown below can be described using the deflection angle θ. Hence it is a single degree of freedom (SDOF) system. θ SDOF system On the other hand, the double pendulum shown below requires at least two deflection angles θ 1 and θ 2 to describe its motion. Hence the system has two DOF's. θ 1 θ 2 MDOF system 22 Downloaded from

23 An introduction to Earthquake engineering Free vibration: A structure is said to be undergoing free vibration when it is disturbed from its static equilibrium position and then allowed to vibrate without any external dynamic excitation (force). Hence a free vibration is initiated by giving the system some initial displacement u(0) and velocity v(0) when time t is zero. 23 Downloaded from

24 An introduction to Earthquake engineering Ch 3.0- Structural Dynamics (part-ii) Single degree of freedom systems (SDOF) Any system having its mass concentrated at one location can be modelled as a SDOF system. A mass-spring -damper system can be used to represent a SDOF system as shown below. SDOF mass-spring-damper system Where, K is elastic stiffness coefficient C is damping coefficient m is mass P(t) is dynamic force u(t) is dynamic displacement For e.g. an equivalent single storey model that can be represented by the mass-springdamper system given above is shown below. (Note: k = k1 + k2 ) single storey shear building Newton's Equation of motion Using Newton's second law of motion, the equation of motion can be derived for the SDOF system. From the equation of motion the displacement response of the system the dynamic loading can be determined. Total applied force - total resistant forces = net force P(t) F D F S = m. a P(t) = m. a + F D + F S m. d2 u du dt2 + c. + k. u = P(t) dt m. u + c. u + k. u = P(t)...(i) equation (i) is the general equation of motion 24 Downloaded from

25 An introduction to Earthquake engineering P(t) is externally applied dynamic force F D is damping force F S is elastic spring force a is acceleration of mass m Undamped free vibration (P(t) = 0, c = 0) For undamped free vibration, the external load P(t) is zero since there is no load acting after the initial displacement is imparted. In addition the damping force F D = c. v is also zero for the undamped case, i.e. damping coefficient, c=0. Also, free vibration is initiated by disturbing the system from its static equilibrium position by imparting the mass some displacement u(0) and velocity u (0) at time zero. Hence, the undamped free vibration equation can be formulated and solved as follows; m. u + c. u + k. u = P(t) m. u + k. u = 0...(i) (since for free vibration, C=0, P(t) =0) The solution to equation (ii) has the form, u = e st, where the constant s is unknown Substituting u = e st into eq(i), we get, ( ms 2 + k)e st = 0, the exponential e st cannot be zero, e st 0, so the characteristic equation is, ms 2 + k = 0 ms 2 + k = 0 s 2 = k m s 2 = ω n 2 ( ω n 2 = k m ) s 1,2 = ±i. ω n (where i = ( 1) The general solution of the equation of motion is, u(t) = A 1 e s1.t + A 2 e s2.t Substituting S1 and S2, u(t) = A 1 e i.ωt + A 2 e i.ωt, where A1 and A2 are constants to be determined Using de Moivre's theorem, x = eix +e ix, sinx = eix e ix 2 2i we have, u(t) = A. cosω n t + B. sinω n t...(ii) A and B are constants to be determined Differentiating eq(ii) we get, u (t) = ω n A. sinω n t + ω n B. cosω n t...(iii) Evaluating eq(ii) and eq(iii) at time t=0, we can determine constants A and B, 25 Downloaded from

26 An introduction to Earthquake engineering from eq(ii), u(t) = u(0) = A ( where u(0) is initial displacement) A = u(0) from eq(iii), u (t) = u (0) = ω n. B (u (0) is the initial velocity) B = u (0) ω n Substituting A & B into eq(ii) gives the solution as, u(t) = A. cosω n t + B. sinω n t u(t) = u(0). cosω n t + u (0) ω n. sinω n t.. (iv) u(0)is initial displacement at t = 0 V(0) is initial velocity t = 0 u 0 is amplitude of free vibration T n is natural period of free vibration ω n is natural frequency of vibration The motion described by equation (iv) is shown in the figure above and is known as a simple harmonic motion. The undamped system oscillates back and forth between the maximum displacement u 0 and minimum displacement u 0. The magnitude u 0 is called the amplitude of the motion, is given by; u 0 = [u(0)] 2 u (0) + [ ] 2 ω n Hence the amplitude u 0 depends on the initial displacement u(0) and the initial velocity u (0) given to the system at time t=0. Damped free vibration response (P(t)=0, C 0) The general governing equation of motion is given by; m. u + c. u + k. u = 0 Dividing the equation by m gives, u + 2ξω n u + ω n 2 u = 0 26 Downloaded from

27 An introduction to Earthquake engineering ξ is damping ratio C cr is criticial damping coefficient ξ = c = c, c c cr 2mω cr = 2mω n = 2 k. m n Based on the damping ratio ξ acting on a system there can be three types of systems, 1. Critically-damped systems (ξ = 1) 2. over-critically damped systems (ξ > 1) 3. Under-critically damped systems (ξ < 1) 1) Critically-damped systems (ξ = 1): When the value of the damping coefficient C is equal to the critical damping coefficient Ccr the system returns to its equilibrium position without oscillating. i.e. if C = C cr, C = 2mω n then the system will be critically damped In terms of damping ratio, ξ = C C cr = 1 Hence critical-damping in a system represents the smallest amount of damping for which no oscillation occurs in the free-vibration response. 2) Over-critically damped systems (over damped) If C >C cr, or ξ > 1 the system does not oscillate and returns to its equilibrium position at a slower rate than a critically damped system. Free vibration response for systems with critical and overcritical damping 3) Under-critically damped systems (underdamped) If the damping coefficient c is less than the critical damping C cr, the system oscillates about its equilibrium position with a progressively decreasing amplitude i.e. if C <C cr, or ξ<1 the system will be underdamped Majority of structures such as buildings, bridges, dams, nuclear power plants, offshore structures etc are underdamped (ξ<1), with their damping ratio typically less than 10% or ξ< Downloaded from

28 An introduction to Earthquake engineering The response for an undercritically (underdamped) free vibration system can be determined as follows; Governing differential equation is, u + 2ξω n u + ω n 2 u = 0...(i) The solution to the differential equation has the form, u = e st Substituting into eq(i), (s 2 + 2ξω n s + ω n 2 )e st = 0, The equation is satisfied for all values of t if, s 2 + 2ξω n s + ω n 2 = 0 (characteristic equation) The characteristic equation has two roots, s 1,2 = ω n ( ξ ± i. 1 ξ 2 ) Hence the general solution is, u(t) = A 1 e s 1t + A 2 e s 2t u(t) = e ξω nt (A 1 e iω Dt + A 2 e iω Dt )...(ii) where A 1 & A 2 are constants to be determined, ω D is the damped natural frequency, ω D = ω n 1 ξ 2 eq (ii) can be written in terms of trigonometric functions, u(t) = e ξωnt (A. cosω D t + B.Sinω D t)...(iii) (A,B are constants to be determined) Using the initial conditions the particular solution can be found, u = u(0), u = u (0) for t = 0, u = u(0) in eq(iii), u(0) = A for t=0, u = u (0) in eq(iii), B = u (0)+ ξω nu(0) ω D Substituting A and B into eq(iii) we get, u(t) = e ξωnt [u(0). cosω D t + u (0)+ ξω nu(0). Sinω ω D t]...(iv) D Equation (iv) gives the response u(t) for an underdamped free vibration system Damping lowers the natural frequency from ω n to ω D, hence lengthens the natural period from T n to T D. T n = 2π ω n, T D = 2π ω D, T D = T n 1 ξ 2 T D is damped natural period ω D is damped natural frequency 28 Downloaded from

29 An introduction to Earthquake engineering eq(iv) indicates that the displacement amplitude decays exponentially with time. The envelope curve is ±ρe ξω nt as shown in figure above. ρ = (u(0)) 2 + ( u (0)+ ξω nu(0) ω D ) 2 Logarithmic decrement The damping in a system can be evaluated from the record of free vibration. The decay of successive amplitudes of motion which is expressed by the logarithmic decrement is related to the damping ratio. The logarithmic decrement for a system is defined as the natural logarithm of any two successive amplitudes in free vibration. From the equation to the exponential curve above, u 1 = ρe ξω nt u 2 = ρe ξω n(t+t D ) The logarithmic decrement is given by, δ = ln ( u 1 u 2 ) ln ( u 1 u 2 ) = ( ω n ξt) [ ω n ξ(t + T D )] = ω n ξt D = ω n ξ ( 2π δ = 2πξ 1 ξ 2 ω D ) If ξ is very small ( ) then 1 ξ 2 1 Hence, logarithmic decrement can be approximated using, δ = 2πξ 29 Downloaded from

30 An introduction to Earthquake engineering Multi-degree of freedom systems (MDOF) MDOF modelling For most civil engineering structures, the dynamic response cannot be described accurately using a SDOF model. Instead, MDOF modelling has to be carried out to obtain a more accurate response of the structural system to a dynamic loading. One of the most common models used for a multi-storey building is the shear building model. The assumptions made in constructing a shear building model are; 1) The total mass of the structure is concentrated at the levels of the floors. 2) The girders on the floors are infinitely rigid as compared to the columns. Hence the floors only undergo only rigid lateral displacement. 3) The deformation of the structure is independent of the axial forces present in the columns. 30 Downloaded from

31 An introduction to Earthquake engineering fig.1 Shear building model Advantages and disadvantages of SDOF and MDOF models SDOF MDOF 1. Modelling cost is low 1. Modelling cost can be high 2. SDOF models are not 2. More accurate than SDOF models accurate for most structures 3. Difficult to assess the reliability of results obtained 3. Greater number of DOF's provide better approximation but using two to three DOF's can also give good results The equations of motion for a MDOF system can be formulated using dynamic equilibrium condition. The four types of forces that need to be considered for dynamic equilibrium at any point are; 1) Externally applied load - P(t) 2) Inertia force - F I 3) Damping force- F D 4) Elastic force- F S Therefore for each degree of freedom (DOF) the dynamic equilibrium can be expressed as; F I1 + F D1 + F S1 = P 1 (t) 31 Downloaded from

32 An introduction to Earthquake engineering F I2 + F D2 + F S2 = P 2 (t) F I3 + F D3 + F S3 = P 3 (t) F IN + F DN + F SN = P N (t) The equilibrium equations can be expressed more conveniently in matrix form as, [F I ] + [F D ] + [F S ] = [P(t)]... eq (i) F I1 F D1 F S1 P 1 (t) F I2 F D2 F S2 P 2 (t) Where, [F I ] =., [F D ] =., [F S ] =., [P(t)] =. { F IN } { F DN } { F SN } { P N (t)} Also, we have, [F I ] = [m]. {u } vector [F D ] = [c]. {u } where, [m] is the mass matrix, {u } is the acceleration [c] is the damping matrix, {u } is the velocity vector vector [F s ] = [k]. {u} [k] is the stiffness matrix, {u} is the displacement Substituting into eq(i), we get [m]. {u } + [c]. {u } + [k]. {u} = {P(t)}... eq(ii) Equation (ii) is the equation of motion for MDOF systems in matrix format The three storey shear building (see fig.1) can also be represented using the MDOF massspring system as shown below; MDOF mass-spring system 32 Downloaded from

33 An introduction to Earthquake engineering The equations of motion can be derived for each mass using Newton's law. Hence, for mass m 1, P 1 (t) k 1 u 1 + k 2 (u 2 u 1 ) m 1 u 1 = 0 m 1 u 1 + k 1 u 1 k 2 (u 2 u 1 ) = P 1 (t) m 1 u 1 + k 1 u 1 k 2 u 2 + k 2 u 1 = P 1 (t) m 1 u 1 + (k 1 + k 2 ). u 1 k 2 u 2 = P 1 (t)...(i) For mass m 2, P 2 (t) k 2 (u 2 u 1 ) + k 3 (u 3 u 2 ) m 2 u 2 = 0 m 2 u 2 + k 2 (u 2 u 1 ) k 3 (u 3 u 2 ) = P 2 (t) m 2 u 2 + k 2 u 2 k 2 u 1 k 3 u 3 + k 3 u 2 = P 2 (t) m 2 u 2 k 2 u 1 + (k 2 + k 3 )u 2 k 3 u 3 = P 2 (t)...(ii) For mass m 3, P 3 (t) k 3 (u 3 u 2 ) m 3 u 3 = 0 m 3 u 3 + k 3 (u 3 u 2 ) = P 3 (t) m 3 u 3 k 3 u 2 + k 3 u 3 = P 3 (t)...(iii) Hence, for masses m 1, m 2, and m 3 from eq(i), (ii), and (iii), m 1 u 1 + (k 1 + k 2 ). u 1 k 2 u 2 = P 1 (t) m 2 u 2 k 2 u 1 + (k 2 + k 3 )u 2 k 3 u 3 = P 2 (t) m 3 u 3 k 3 u 2 + k 3 u 3 = P 3 (t) Organizing the three equations in matrix format, m u 1 (k 1 + k 2 ) k 2 0 P 1 (t) [ 0 m 2 0 ]. { u 2} + [ k 2 (k 2 + k 3 ) k 3 ]. { u 2 } = { P 2 (t)} 0 0 m 3 u 3 0 k 3 k 3 u 3 P 3 (t) u 1 or, [m]. {u } + [k]. {u} = {P(t)} where, [m] is the mass matrix, {u } is the acceleration vector [k] is the stiffness matrix, {u} is the displacement vector 33 Downloaded from

34 An introduction to Earthquake engineering {P(t)} is force vector Free vibration analysis of undamped MDOF systems (P(t) = 0) The equations of motion for an undamped MDOF system under free vibration is given by, [m]. {u } + [k]. {u} = {0}...(i) eq(i) can be expressed as, ([k]. ω 2 [m]). {φ} = {0}...(ii) Eq (ii) is known as an eigenvalue problem. The quantities ω 2 are the eigenvalues or square of free-vibration natural frequencies, while the corresponding displacement vectors φ express the corresponding shapes of the vibrating system and are called eigenvectors or mode shapes. The non-trivial solution to eq(ii) is given by, [k]. ω 2 [m] = 0 Expanding the determinant will give an algebraic equation of nth degree for the frequency parameter ω 2 for a system having n degrees of freedom. The n roots of this equation (ω 1, ω 2, ω 3,.., ω n ) represent the frequencies of n modes of vibration which are possible in the system. The mode having the lowest frequency is called the first mode or fundamental mode. The next higher frequency is second mode etc. The vector made of the entire set of modal frequencies, arranged in sequence, is called the frequency vector {ω} ω 1 {ω} = ω. 2. { ω n } The mode shape vector {φ} represents the shape of the vibrating system for various modes. {φ} is a dimensionless vector which is expressed for mode n as ; {φ} = φ 1n φ 2n.. where, n is mode number, N is no of DOF's { φ Nn } 34 Downloaded from

35 An introduction to Earthquake engineering Chapter 4 Lateral load resisting systems for buildings Lateral force-resisting systems A structural system must be selected by the designer to resist the lateral loading that my act on the building. Aspects of structural configuration, symmetry, mass distribution, and vertical regularity may need to be discussed between the architect and the structural engineer. Important criteria such as adequate strength, stiffness, and ductility must also be considered for the building to have satisfactory response during an earthquake. The lateral load in a building is transferred as follows; Horizontal seismic inertia forces are developed in the structure when subjected to earthquake loading. These inertial forces are generated at the various floor levels. The floor and roof slabs are also called diaphragms and have high in-plane rigidity when made of concrete (RCC) The diaphragms transfer the inertial load to the vertical load resisting elements. In the case of wall systems, the shear walls are the primary vertical members which resist most of the lateral loading from the diaphragms. In the case of RC frame the columns resist the lateral loading. The vertical elements transfer the loads to the foundation and hence the load path is completed at the bearing soil underneath the foundation. Hence, the floor and roof diaphragms, along with the shear walls and columns together work as the lateral load resisting system in a building. A well-designed and well-built building has a reliable load path, established by design. This load path transfers the lateral forces over the full height of the building from the roof to the foundation. Lateral force resisting systems can generally be divided on the basis of structural systems 1) Structural frame system 2) Structural wall system 3) Dual frame-wall systems These three categories can be further subdivided as given in table 7 of IS1893:part1 (2002). 1. Structural frame system Structures of multi-storey reinforced concrete buildings often consist of frames. Beams, supporting floors, and columns are continuous and meet at nodes, often called rigid 35 Downloaded from

36 An introduction to Earthquake engineering joints. Such frames are capable of carrying gravity loads while providing adequate resistance to horizontal forces, acting in any direction. 2. Structural wall system Structural walls are the key structural components that resist the lateral forces due to earthquake in a load-bearing masonry building. Gravity load effects on masonry structural walls are usually not of significance considering design. Hence, if necessary, only the lateral load resistance of shear walls may have to be checked during design. A shear wall system has lower redundancy and has less inelastic response capacity. Hence in IS1893 the response reduction factor for a shear wall system is less when compared to a frame or dual system. 3. Dual System This system consists of reinforced concrete frames and RC or masonry walls, interacting together to provide the required resistance to lateral forces. Each structural system also carries its appropriate share of gravity load in proportion to its lateral stiffness. Dual system is also known as hybrid or wall-frame system. A dual system is also designed so that the moment frame independently resists at least 25% of the design base shear due to the earthquake. A dual system is more redundant and hence has a higher value for response reduction factor 'R' in IS1893. Effects of irregularities on buildings A building that lacks symmetry and has discontinuity in geometry, mass, or load resisting elements is called as irregular building. The structure should have a robust and continuous lateral load path in order to safely transfer the seismic forces to the ground. In general structural irregularities can be grouped into vertical irregularities and plan irregularities. 1. Vertical irregularities: stiffness irregularity mass irregularity vertical geometric irregularity 2. Plan irregularities: Torsional irregularity Re-entrant corners Non-parallel systems diaphragm discontinuity 36 Downloaded from

37 An introduction to Earthquake engineering Vertical irregularity Vertical irregularity occurs due to sudden and significant change of strength, stiffness, geometry, and mass over the height of a building. Stiffness irregularity: According to IS-1893(2002) a soft storey is that in which the lateral stiffness is less than 70% the lateral stiffness of the storey above or less than 80% of the average lateral stiffness of the three stories above. Extreme soft storey is defined as having a lateral stiffness less than 60% the lateral stiffness of the storey above or less than 70% of the average lateral stiffness of the three storeys above. The upper storey moves as a single rigid unit and hence accumulative deformation due to earthquake lateral loading occurs mainly in the ground storey. The soft storey can fail if plastic hinges are formed on the soft storey columns. Mass irregularity 37 Downloaded from

38 An introduction to Earthquake engineering From IS 1893(2002) mass irregularity can be said to exist where the seismic weight of any storey is more than 200% of that of its adjacent storeys. The seismic weight a floor is its full dead load plus a percentage of its imposed load as specified in the code. Vertical geometric irregularity It can be considered to exist where the horizontal dimension of the lateral force-resisting system in any storey is more than 150% of that in its adjacent storey. Vertical discontinuities in load path A discontinuous load path in a structure can contribute significantly to structural damage during an earthquake. Structural elements that are part of the load path must all be tied together in order to have a load path that efficiently transfers the lateral seismic forces to the ground. Irregularities in load path includes discontinuous columns, shear walls, bracing, and frames. 38 Downloaded from

39 An introduction to Earthquake engineering Lateral strength irregularity: A weak storey is one in which the storey lateral strength is less than 80% if that in the storey above. The storey lateral strength is the total strength of all seismic force-resisting elements sharing the storey shear in considered direction. Rigid Floor diaphragms Rigid floor diaphragms have high in-plane stiffness compared to vertical members such as columns and walls. Hence rigid diaphragms are assumed to be infinitely rigid for in-plane bending, which means that the diaphragm will undergo only rigid body translation in the horizontal x-y plane and rigid body rotation in the vertical z-axis. Rigid diaphragm distributes the horizontal forces to the vertical resisting elements in direct distribution to their relative rigidities (stiffness). This is based on the assumption that the diaphragm does not deform itself and will cause each vertical element to deflect the same amount. Reinforced concrete slabs and composite steel deck are rigid diaphragms, whereas timber flooring is a flexible diaphragm. Center of mass: It is the point where the entire mass of a system can be thought to be concentrated. The resultant of the earthquake force acts through the center of mass. For a system with uniform density the center of mass is located at its geometric center (centroid). 39 Downloaded from

40 An introduction to Earthquake engineering During an earthquake, acceleration induced inertia forces will be developed at each floor level, where the mass of the entire storey may be assumed to be located. The resultant of the lateral inertial force will act through the center of mass of the floor. In regular buildings the positions of center of floor masses will differ very little from floor to floor. However, irregular mass distribution over the height of the building may result in variations in location of center of masses. Center of rigidity ( center of stiffness): It is the point through which the resultant of the resisting forces developed in the system passes. Hence, center of rigidity is also the point on the diaphragm where the application of lateral force will cause only rigid body translation and no rigid body rotation. Moment resisting frames (MRF) Any frame having a beam with a finite stiffness (EI) value and a rigid beam-column joint can be called a moment resisting frame. A moment resisting frame resists lateral earthquake forces primarily by flexure. The joints between the beams and columns are designed to be rigid. The rigid joint in the frame maintains a constant angle (90 ) between the beam and column under load application. This rigid joint ensures that the ends of beams and columns meeting joints must rotate by same amount. However, since the beam is infinitely rigid, it cannot deform and hence, the joint itself is not allowed to rotate. In this case, the bending moment in the column is reduced to half of that frame without rigid (flexible) joint. Inelastic deformability is achieved by specially detailing the beams, columns, and the beam-column joints. The structural detailing in moment resisting frames must ensure that ductile failure of frame members occur before shear or brittle failure. The lateral stiffness of moment frames are comparatively less than the shear wall and braced systems. Hence, the lateral deflections can be relatively large for the frame system. 40 Downloaded from