Seismic Analysis of Structures - I


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Seismic Analysis of Structures - I

  1. 1. T.K. DattaDepartment Of Civil Engineering, IITChapters – 1 & 2Chapter -1SEISMOLOGY
  2. 2. T.K. DattaDepartment Of Civil Engineering, IITIntroduction It is a big subject and mainly deals withearthquake as a geological process. However, some portions of seismology areof great interest to earthquake engineers. They include causes of earthquake, earthquakewaves, measurement of earthquake, effect ofsoil condition on earthquake, earthquake pre-diction and earthquake hazard analysis Understanding of these topics help earthquakeengineers in dealing seismic effects on structuresin a better way. Further knowledge of seismology is helpful indescribing earthquake inputs for structures whereenough recorded data is not available.Lec-1/1Seismology
  3. 3. T.K. DattaDepartment Of Civil Engineering, IITInteriors of earth184 −− kmstoVpLec-1/2 Before earthquake is looked as a geologicalprocess, some knowledge about the structure ofearth is in order.In-side the earth Crust: 5-40 km;M discontinuity; floating Mantle: lithosphere (120 km);asthenosphere-plasticmolten rock (200 km);bottom- homogenous;variation of v is less(1000 km - 2900 km)Core: discovered by Wichert &Oldham; only P waves canpass through inner core(1290 km); very dense;nickel & iron; outer core(2200 km), same density;25000C; 4x106atm;14 g/cm3 Lithosphere floats as a cluster of plates withdifferent movements in different directions.Fig 1.1Seismology
  4. 4. T.K. DattaDepartment Of Civil Engineering, IITPlate tectonics At mid oceanic ridges, twocontinents which were joinedtogether drifted apart due toflow of hot mantle upward. Flow takes place because ofconvective circulation ofearths mantle; energy comesfrom radioactivity inside theearth. Hot material cools as it comesup; additional crust is formedwhich moves outward.Lec-1/3Convective currents Concept of plate tectonics evolved fromcontinental drift.Fig 1.2Seismology
  5. 5. T.K. DattaDepartment Of Civil Engineering, IITContd... New crust sinks beneath sea surface; spreadingcontinues until lithosphere reaches deep seatrenches where subduction takes place. Continental motions are associated with a varietyof circulation patterns. As a result, motions take place through sliding oflithosphere in pieces- called tectonic plates. There are seven such major tectonic plates andmany smaller ones. They move in different directions at differentspeeds.Lec-1/4Seismology
  6. 6. T.K. DattaDepartment Of Civil Engineering, IITContd...Lec-1/5Fig 1.3Major tectonic platesSeismology
  7. 7. T.K. DattaDepartment Of Civil Engineering, IIT Three types of Inter plate interactions exist givingthree types of boundaries.Contd...Lec-1/6 Tectonic plates pass each other at the transformfaults.Fig 1.4Types of interplate boundariesSeismology
  8. 8. T.K. DattaDepartment Of Civil Engineering, IIT Faults at the plate boundaries are the likelylocations for earthquakes - inter plate earth-quake. Earthquakes occurring within the plate arecaused due to mutual slip of rock bedreleasing energy- intra plate earthquake. Slip creates new faults, but faults are mainlythe causes rather than results of earthquake. At the faults two different types of slipageare observed- Dip slip; Strike slip. In reality combination of the types of slipageis observed at the fault line.Contd...Lec-1/7Seismology
  9. 9. T.K. DattaDepartment Of Civil Engineering, IITContd...Lec-1/8Types of faultFig 1.5Seismology
  10. 10. T.K. DattaDepartment Of Civil Engineering, IITCauses of earthquake There are many theories to explain causes ofearthquake. Out of them, tectonic theory of earthquake ispopular. The tectonic theory stipulates that movementsof tectonic plates relative to each other lead toaccumulation of stresses at the plate boundar-ies & inside the plate. This accumulation of stresses finally results ininter plate or intra plate earthquakes. In inter- plate earthquake the existing faultlines are affected while intra-plate earthquakenew faults are created.Lec-2/1Seismology
  11. 11. T.K. DattaDepartment Of Civil Engineering, IITContd... During earthquake, slip takes place at the fault;length over which slip takes place could be severalkilometres; earthquake origin is a point that movesalong the fault line. Elastic rebound theory, put forward by Reid, givescredence to earthquake caused by slip alongfaults. Large amplitude shearing displacement that tookplace over a large length along the San andreasfault led to elastic rebound theory. Modelling of earthquake based on elastic reboundtheory is of two types: Kinematic-time history of slip Dynamic-shear crack and its growthLec-2/2Seismology
  12. 12. T.K. DattaDepartment Of Civil Engineering, IITContd...Fault LineAfter earthquakeDirection of motionDirection of motionRoadFault LineBefore StrainingDirection of motionDirection of motionFault LineStrained (Before earthquake)Direction of motionDirection of motionRoadLec-2/3Fig 1.6Seismology
  13. 13. T.K. DattaDepartment Of Civil Engineering, IITContd… An earthquake caused by slip at the fault proceeds inthe following way: Owing to various slow tectonic activities,strains accumulate at the fault over a longtime. Large field of strain reaches limiting value atsome point of time. Slip occurs due to crushing of rock& masses;the strain is released, releasing vast energyequivalent to blasting of several atom bombs. Strained layers of rock masses bounces backto its unstrained condition.Lec-2/4Seismology
  14. 14. T.K. DattaDepartment Of Civil Engineering, IITContd...FaultBefore slip Rebound due to slipPush and pull force Double coupleLec-2/5Fig 1.7Slip could be of any type-dip, strike or mixed givingrise to a push & pull forcesacting at the fault; slipvelocity at an active fault-10to 100mm/year.This situation is equivalentto two pairs of coupledforces suddenly acting andthus, moving masses ofrock leading to radialwaves propagating in alldirections.Seismology
  15. 15. T.K. DattaDepartment Of Civil Engineering, IITContd… Propagating wave is complex& is responsiblefor creating displacement and acceleration ofsoil/rock particle in the ground. The majority of the waves travels through therocks within the crust and then passes throughthe soil to the top surface. Other theory of tectonic earthquake stipulatesthat the earthquake occurs due to phasechanges of rock mass, accompanied by volumechanges in small volume of crust. Those who favour this theory argues thatearthquakes do occur at greater depthswhere faults do not exist.Lec-2/6Seismology
  16. 16. T.K. DattaDepartment Of Civil Engineering, IITSeismic waves Large strain energy released during earthquakepropagates in all directions within earth as elasticmedium. These waves, called seismic waves, transmitenergy from one point to the other & finally carryit to the surface. Within earth, waves travel in almost homogeno-us elastic unbounded medium as body waves. On the surface, they move as surface waves. Reflection & refraction of waves take place nearthe surface at every layer; as a result waves getmodified.Lec-2/7Seismology
  17. 17. T.K. DattaDepartment Of Civil Engineering, IITContd... Body waves are of two types- P & S waves;S waves are also called transverse waves. Waves propagation velocities are given by: P waves arrive ahead of S waves at a point; timeinterval is given by: Polarized transverse waves are polarization of particl-es either in vertical(SV) or in horizontal(SH) plane.( )( )( ))2.1(121)1.1(21112/12/12/1+==−+−=νρρννννρνEGEsp)3.1(11−∆=pspTννLec-2/8Seismology
  18. 18. T.K. DattaDepartment Of Civil Engineering, IITSurface waves are of two types - L wavesand R waves. L waves: particles move in horizontal planeperpendicular to the direction of wavepropagation. R waves:- particles move in vertical plane;they trace a retrogate elliptical path; foroceanic waves water particles undergosimilar elliptical motion in ellipsoid surfaceas waves pass by. L waves move faster than R waves onthe surface (R wave velocity ~0.9 )Contd...SVLec-2/9Seismology
  19. 19. T.K. DattaDepartment Of Civil Engineering, IITContd...Lec-2/10Body & Surface wavesFig 1.8Seismology
  20. 20. T.K. DattaDepartment Of Civil Engineering, IIT P& S waves change phases as PPP, PS, PPSetc. after reflection & refraction at the surface.Contd...PSPSSSPPSSPPLec-2/11Reflection at the earth surfaceFig 1.9Seismology
  21. 21. T.K. DattaDepartment Of Civil Engineering, IITRecords of surface waves Strong earthquake waves recorded on the surfaceare irregular in nature.P PP S SS L They can generally be classified in four groups: Practically Single Shock: near source; on firmground; shallow earthquake. Moderately long irregular: moderate distancefrom source; on firm ground-elcentro earthquake.Lec-2/12Typical strong motion recordFig 1.10Seismology
  22. 22. T.K. DattaDepartment Of Civil Engineering, IIT A long ground motion with prevailing period:filtered ground motion through soft soil,medium- Loma Prieta earthquake. Ground motion involving large Scale groundDeformation: land slides, soil liquefaction-Chilean & Alaska earthquakes.Contd..Lec-2/13 Most ground motions are intermediate betweenthose described before (mixed). Amongst them, nearly white noise type earth-quake records ( having a variety of frequencycompositions are more frequent on firm ground ).Seismology
  23. 23. T.K. DattaDepartment Of Civil Engineering, IIT0. (sec)0.5 1.0 1.5 2(a)AccelerationContd...Lec-2/14Single Shock10.01WESTEASTDisplacement(cm)Time (sec)0.5 1.0 1.5 2displacementFig 1.11Seismology
  24. 24. T.K. DattaDepartment Of Civil Engineering, IITContd..0 5 10 15 20 25 30-0.4- (sec)Acceleration(g)AccelerationLec-2/15Mixed frequency0 5 10 15 20 25 30-10-5051015Time (sec)Displacement(cm)DisplacementFig 1.12Seismology
  25. 25. T.K. DattaDepartment Of Civil Engineering, IITContd..0 1 2 3 4 5 6 7 8-6-4-2024Time (sec)Displacement(cm)DisplacementLec-2/16Time(sec)0 2 4 6 8 10 12 14 16 18-0.5-0.4-0.3-0.2- frequencyFig 1.13Seismology
  26. 26. T.K. DattaDepartment Of Civil Engineering, IIT They refer to quantities by which size & energyof earthquakes are described. There are many measurement parameters; someof them are directly measured; some areindirectly derived from the measured ones. There are many empirical relationships that aredeveloped to relate one parameter to the other. Many of those empirical relationships and theparameters are used as inputs for seismicanalysis of structures; so they are describedalong with the seismic inputs.Earthquake measurement parametersLec-3/1Seismology
  27. 27. T.K. DattaDepartment Of Civil Engineering, IIT Here, mainly two most important parameters,magnitude & intensity of earthquake are describedalong with some terminologies.Contd... Most of the damaging earthquakes haveEpicentre Epicentral DistanceHypocentral DistanceFocal DepthFocus/HypocentreSite Limited region of earthinfluenced by the focusis called focal region ;greater the size ofearthquake, greater isthe focal region. shallow focal depth <70 km; depths of foci >70 km are intermediate/deep.Lec-3/2Earthquake definitionsFig 1.14Seismology
  28. 28. T.K. DattaDepartment Of Civil Engineering, IITContd... Force shocks are defined as those which occurbefore the main shock. After shocks are those which occur after the mainshock. Magnitude of earthquake is a measure of energyreleased by the earthquake and has the followingattributes: is independent of place of observation. is a function of measured maximum displace-ments of ground at specified locations. first developed by Waditi & Richter in 1935.Lec-3/3Seismology
  29. 29. T.K. DattaDepartment Of Civil Engineering, IITContd... magnitude (M) scale is open ended. M > 8.5 is rare; M < 2.5 is not perceptible. there are many varieties of magnitude ofearthquake depending upon waves andquantities being measured. Local magnitude ( ), originally proposed byRichter, is defined as log a (maximum amplitudein microns); Wood Anderson seismograph:R=100 km; magnification: 2800:LMpT = 0.8s :ξ = 0.8)6.1(log7.248.2log ∆+−= AMLLec-3/4Seismology
  30. 30. T.K. DattaDepartment Of Civil Engineering, IIT Since Wood Anderson seismograph is no more inuse, coda length ( T ), defined as total signalduration, is used these days: Body magnitude ( ) is proposed by Gutenberg& Richter because of limitations of instrument &distance problems associated with . It is obtained from compression P waves withperiods in the range of 1s; first few cycles areused;Contd...)7.1(logTbaM L +=bMLM( ) )8.1(,log ∆+= hQTAMbLec-3/5Seismology
  31. 31. T.K. DattaDepartment Of Civil Engineering, IITOccasionally, long period instruments are usedfor periods 5s-15s.Surface magnitude ( ) was again proposed byGutenberg & Richter mainly for largeepicentral distance.However, it may be used for any epicentraldistance & any seismograph can be used.Praga formulation is used with surface waveperiod of the order of 20sA is amp of Rayleigh wave (20s); is in km.sMContd...)9.1(0.2log66.1log +∆+=TAMs∆Lec-3/6Seismology
  32. 32. T.K. DattaDepartment Of Civil Engineering, IIT Seismic moment magnitude ( ) is a bettermeasure of large size earthquake with the helpof seismic moment.A- area (m²) ; U- longitudinal displacement(m);G(3x10¹ºN/m²). Seismic Moment ( ) is measured fromseismographs using long period waves anddescribes strain energy released from the entirerupture surface.wMContd...( 1.10)oM GUA=oMLec-3/7 Kanamori designed a scale which relates to.wMoMSeismology
  33. 33. T.K. DattaDepartment Of Civil Engineering, IITContd...2 3 4 5 6 7 8 9 1023456789MLMsMsMJMAMBMLMbM~MWMoment Magnitude MwMagnitude)11.1(0.6log3210 −= ow MMLec-3/8Fig 1.15Seismology
  34. 34. T.K. DattaDepartment Of Civil Engineering, IIT Energy Release, E ( Joules ) is given by :M(7.3) ~ 50 megaton nuclear explosionM(7.2) releases 32 times more energy thanM(6.2)M(8) releases 1000 times more energy thanM(6) Some Empirical formulae [L (km); D/U(m);A(km2)]Contd...sME 158.410 +=)14.1()42.0(46.582.0)14.1()24.0(49.391.0)14.1()22.0(22.369.0)14.1(27.4)log32.1()13.1(65.5)log98.0(dMLogDcMLogAbMLogLaUMLMLogDwLogAwLogLw=−==−==−=+=+=σσσLec-3/9Seismology
  35. 35. T.K. DattaDepartment Of Civil Engineering, IIT Intensity is a subjective measure of earthquake;human feeling; effects on structures; damages. Many Intensity scales exist in different parts of theworld; some old ones: Gastaldi Scale (1564) Pignafaro Scale(1783) Rossi- forel Scale(1883) Mercalli – Cancani – Sieberg scale is still in useinwestern Europe. Modified Mercalli Scale (12 grade) is widelyusednow.Contd...Lec-3/10Seismology
  36. 36. T.K. DattaDepartment Of Civil Engineering, IITContd...Lec-3/11Intensity Evaluation DescriptionMagnitude(Richter Scale)I Insignificant Only detected by instruments 1- – 1.9II Very LightOnly felt by sensitive persons; oscillation ofhanging objects2 – 2.9III Light Small vibratory motion 3 – 3.9IV ModerateFelt inside building; noise produced bymoving objects4 – 4.9V Slightly StrongFelt by most persons; some panic; minordamagesVI StrongDamage to non-seismic resistancestructures5 – 5.9VII Very StrongPeople running; some damages in seismicresistant structures and serious damage toun-reinforced masonry structuresVIII Destructive Serious damage to structures in generalIX RuinousSerious damage to well built structures;almost total destruction of non-seismicresistant structures6 – 6.9X DisastrousOnly seismic resistant structures remainstanding7 – 7.9XIDisastrous inExtremeGeneral panic; almost total destruction; theground cracks and opensXII Catastrophic Total destruction 8 – 8.9Seismology
  37. 37. T.K. DattaDepartment Of Civil Engineering, IIT There have been attempts to relate subjectiveintensity with the measured magnitude resultingin several empirical equations: Other important earthquake measurementparameters are PGA, PGV, PGD. PGA is more common & is related to magnitudeby various attenuation laws (described in seismicinputs).Contd...max1.3 0.6 (1.15)8.16 1.45 2.46ln (1.16)1.44 ( ) (1.17)sM II M rI M f r= += + −= +Lec-3/12Seismology
  38. 38. T.K. DattaDepartment Of Civil Engineering, IITMeasurement of earthquake Principle of operation is based on the oscillation of apendulum.Lec-4/1Sensor : mass; string;magnet &supportRecorder : drum; pen;chart paperAmp : optical / electro-magnetic meansDamp : electromagnetic/fluid dampersFig 1.16Seismology
  39. 39. T.K. DattaDepartment Of Civil Engineering, IITuHorizontal pendulumVertical pendulumuContd...Lec-4/2Fig 1.17Seismology
  40. 40. T.K. DattaDepartment Of Civil Engineering, IIT Equation of motion of the bob is If T  very large (Long period seismograph) If T  very small (short period seismograph) If T  very close to & 2k  very LargegTContd...22 (1.18)x kx w x uν+ + =−&& && &&)19.1(uxorux ∝−= ν)20.1(2uxoruxw &&&& ∝−= ν(1.21)x u or x uν=− µ& && &Lec-4/3Seismology
  41. 41. T.K. DattaDepartment Of Civil Engineering, IITContd..NS Horseshoe MagnetSuspensionCopper MassMirrorLight Beam copper cylinder2mm / 25mm /0.7g taut wire 0.02mm reflection of beammagnified by 2800 electro - magneticdamping 0.8Lec-4/4Wood Anderson SeismographFig 1.18Seismology
  42. 42. T.K. DattaDepartment Of Civil Engineering, IIT Commonly used seismograph measuresearthquake within 0.5-30 seconds. Strong motion seismograph has the followingcharacteristics:Contd..• period & damping of the pickup of 0.06- 25cps ;• preset acceleration 0.005g;• sensitivity 0.001-1.0g;• average starting time 0.05-0.1s.Lec-4/5Seismology
  43. 43. T.K. DattaDepartment Of Civil Engineering, IIT Local Soil condition may have significant influenceon ground motions. Most of seismic energy at a site travels upwardthrough soil from the crust/rock bed below in theform of S/P waves. In the process, amplitude, frequency contents &duration of earthquake get changed. The extent depends upon geological, geographicaland geotechnical conditions. Most influencing factors are properties of thesoil and topography.Modification of ground motionLec-4/6Seismology
  44. 44. T.K. DattaDepartment Of Civil Engineering, IIT Analysis of collected data revealed interestingfeatures of soil modification:Contd... Attenuation of ground motion through rockbed is significant 0.03g-350km (M=8.1). For very soft soil, predominant period ofground motion changes to soil period; forrock bed PGA 0.03g  (AF=5). Duration increases also for soft soil. Over a loose sandy soil underlying bymud, AF=3 for 0.035g-0.05g (at rock bed). The shape of the response spectrumbecomes narrow banded for soft soil.Lec-4/7Seismology
  45. 45. T.K. DattaDepartment Of Civil Engineering, IITContd... As PGA at the rock bed increases, AFdecreases. For strong ground shaking, PGA amplification islow because of hysteretic behaviour of soil. At the crest of narrow rocky ridge, increasedamplification occurs; AF ≈ 2π/ǿ ( theoreticalanalysis ). At the central region of basin, ID wave propagationanalysis is valid; near the sides of the valley, 2Danalysis is to be carried out. 1D, 2D or 3D wave propagation analysis is carriedout to find PGA amplification theoretically.Lec-4/8Seismology
  46. 46. T.K. DattaDepartment Of Civil Engineering, IITSeismic hazard analysisIt is a quantitative estimation of most possibleground shaking at a site.The estimate can be made using deterministicor probabilistic approaches; they requiresome/all of the following: Knowledge of earthquake sources, fault activity,fault rupture length. Past earthquake data giving the relationshipbetween rupture length & magnitude. Historical & Instrumentally recorded groundmotion.Possible ground shaking may be representedby PGA, PGV, PGD or response spectrumordinates.Lec-4/9Seismology
  47. 47. T.K. DattaDepartment Of Civil Engineering, IIT Deterministic Hazard Analysis (DSHA):A simple procedure to compute groundmotion to be used for safe design ofspeciality structures. Restricted only when sufficient data isnot available to carry out PSHA. It is conservative and does not providelikely hood of failure. It can be used for deterministic design ofstructures. It is quiet often used for microzonation oflarge cities for seismic disaster mitigation.Contd…Lec-4/10Seismology
  48. 48. T.K. DattaDepartment Of Civil Engineering, IITContd…)25ln(80.1859.074.6PGA(gals)ln +−+= rmLec-4/11 It consists of following 5 steps: Identification of sources including their geometry. Evaluation of shortest epicentral distance / hypocentral distance. Identification of maximum likely magnitude ateach source. Selection of the predictive relationship valid forthe region.Seismology
  49. 49. T.K. DattaDepartment Of Civil Engineering, IIT Example 1.1 :Maximum magnitudes forsources 1, 2 and 3 are 7.5,6.8 and 5 respectively.Contd…(-50, 75)Source 1(-15, -30)(-10, 78)(30, 52)(0, 0)Source 3Source 2SiteSources of earthquakenear the site (Examp. 1.1)Source m r(km) PGA1 7.5 23.70 0.490 g2 6.8 60.04 0.10 g3 5.0 78.63 0.015 gHazard level is 0.49g for the siteLec-4/12Fig 1.19Seismology
  50. 50. T.K. DattaDepartment Of Civil Engineering, IIT Probabilistic seismic hazard analysis (PSHA). It predicts the probability of occurrence of acertain level of ground shaking at a site byconsidering uncertainties of: Size of earthquake Location Rate of occurrence of earthquake Predictive relationshipContd…Lec-5/1PSHA is carried out in 4 steps.Seismology
  51. 51. T.K. DattaDepartment Of Civil Engineering, IIT Step 1 consists of following: Identification & characterization ofsource probabilistically. Assumes uniform distribution of pointof earthquake in the source zone . Computation of distribution of rconsidering all points of earthquake aspotential source.Contd…Lec-5/2 2 step consists of following: Determination of the average rate atwhich an earthquake of a particular sizewill be exceeded using G-R recurrencelaw.)23.1()exp(10 ambmam βαλ −== −Seismology
  52. 52. T.K. DattaDepartment Of Civil Engineering, IIT Using the above recurrence law & specifyingmaximum & minimum values of M, followingpdf of M can be derived (ref. book) 3rd step consists of the following: A predictive relationship is used to obtainseismic parameter of interest (say PGA) forgiven values of m , r .Contd…)26.1()]([exp1)]([exp)(0max0mmmmmfM−−−−−=βββLec-5/3 Uncertainty of the relationship is consideredby assuming PGA to be log normally distributed;the relationship provides the mean value; astandard deviation is specified.Seismology
  53. 53. T.K. DattaDepartment Of Civil Engineering, IITContd…Lec-5/4 4th step consists of the following: Combines uncertainties of location, size& predictive relationship by A seismic hazard curve is plotted as(say is PGA level ).)27.1()()(],|[1∫∫∑ >==drdmrfmfrmyYP RiMiNiiySγλyvsyλy By including temporal uncertainty of earthquake(uncertainty of time) in PSHA & assuming it to be aPoisson process, probability of exceedance of thevalue of , of the seismic parameter in T yearsis given by (ref. book)y[ ] 1 (1.28 )y TtP y y e dλ−> = −Seismology
  54. 54. T.K. DattaDepartment Of Civil Engineering, IITExample 1.2 :For the site shown in Fig 1.20,show a typical calculation forPSHA ( use Equation 1.22with σ = 0.57)Contd…(-50,75)Source 1(-15,-30)(0,0)Source 3Source 2Site(5,80)(25,75) (125,75)(125,15)(25,15)Source Recurrence Law Mo MuSource 1 4 7.7Source 2 4 5Source 3 4 7.3mm −=4logλmm 2.151.4log −=λmm 8.03log −=λLec-5/5Fig 1.20Seismology
  55. 55. T.K. DattaDepartment Of Civil Engineering, IITSolution:Location Uncertainty 1st sourceLine is divided in 1000 segments 2nd sourceArea is divided in 2500 parts (2x 1.2)minmin90.1223.72( interval( ) 10)r kmr divide n== =)10(32.3098.145minmax===nrkmrContd…Lec-5/6Seismology
  56. 56. T.K. DattaDepartment Of Civil Engineering, IIT 3rd source :min maxr r r= =Contd…0.00.427.0433.6840.3249.9653.6060.2466.8873.5280.1686.80P[R=r]Epicentral distance, r (km)[R=r]Epicentral distance, r (km)0.0 102030405060708090100P[R=r]Epicentral distance, r (km)1.0Lec-5/7Fig 1.21Fig 1.23Fig 1.22Seismology
  57. 57. T.K. DattaDepartment Of Civil Engineering, IITSize Uncertainty :631.010501.01011048.03342.15.424141======×−×−×−γγγContd…)29.1()(2)(][12212121ammmmfdmmfmmmPmmmM− +==<< ∫Lec-5/8For each source zoneFor source zone 1, mu and m0 are divided in 10divisions.Seismology
  58. 58. T.K. DattaDepartment Of Civil Engineering, IIT Histogram of M for each source zone are shownContd…0.00.8Magnitude, m0.[M=m]4.837.144.174.505.165.495.826.156.486.810., m0.[M=m]0.00.8Magnitude, m0.[M=m]4.837.144.174.505.165.495.826.156.486.81Lec-5/9Fig 1.24Fig 1.25Fig 1.26Seismology
  59. 59. T.K. DattaDepartment Of Civil Engineering, IIT Say, Probability of exceedance of 0.01g is desiredfor m = 4.19, r = 27.04 km for source zone1The above probability is given asContd..[ ]951.0)(165.1)(104.27,19.4|01.0=−−=−===>ZFzZFrmgPGAPzz[ ][ ] [ ] 176.004.2719.404.27,19.4|01.004.27&19.4101.001.0=====>===rPmPrmgPGAPisrmforggγλλLec-5/10[ ][ ] 336.004.27551.019.4====rPmPSeismology
  60. 60. T.K. DattaDepartment Of Civil Engineering, IIT For different levels of PGA, similar values ofcan be obtained. Plot of vs. PGA gives the seismic hazardcurve.λλContd... for other 99 combinations of m & r canobtained & summed up; for source zones 2 & 3,similar exercise can be done; finally,0.01gλ301.0201.0101.001.0 ||| sourgsourgsourgg λλλλ ++=Lec-5/11Seismology
  61. 61. T.K. DattaDepartment Of Civil Engineering, IITContd…Lec-5/12Example-1.3:The seismic hazard curve for a region shows that the annualrate of exceedance of an acceleration 0.25g due toearthquakes (event) is 0.02.What is the prob. that exactlyone one such event and at least one such event will takeplace in 30 years? Also, find that has a 10% prob. ofexceedancein 50 yrs.Solution:Equation 1.28c (book) can be written as%2.451)1()(%333002.0)1()(3002.03002.0=−=≥=×===×−×−−eNPiieteNPi tλλ[ ] [ ] 0021.0501.01ln)1(1ln=−=≥−=tNPλλSeismology
  62. 62. T.K. DattaDepartment Of Civil Engineering, IIT Seismic risk at a site is similar to that of seismichazard determined for a site. It is defined as: P( ) during a certain period (usually 1year). Inverse of risk becomes return period for . The study of seismic risk requires: Source mechanism parameters – focal depth;orientation of faults etc. Recurrence relationship which is used to findPDF. Attenuation Relationships.s ix x≥ixSeismic risk at a siteLec-5/13Seismology
  63. 63. T.K. DattaDepartment Of Civil Engineering, IIT Using the above Information, seismic risk canbe calculated with the help of either Cornellsapproach or Milne & Davenport approach. Using the concept, many empirical equations areobtained with the help of data / informationfor regions. For a particular region, these empiricalequations are developed; for other regions, theymay be use by choosing appropriate values forthe parameters. Some equations are given in the following Many others are given in the book.Contd..Lec-5/14Seismology
  64. 64. T.K. DattaDepartment Of Civil Engineering, IITContd...[ ]( )( )( )1111 111.541( )1 1 1 ( )1 1( )1( ) ( / ) (1.30)exp exp ( ) (1.32 )ln (1.32 )47 (1.33)1( ) | (1.37)11 ( ) (1.38)1 (1.39)os u osops soism MM s o u m Ms Mm MsN Y Y cp m aT bP I i eeF m P M m M m MeP M m F mP M m eβββα βα α−−− −− −− −== − −=≥ =− = ≤ ≤ ≤ =  −≥ = −≥ = −Lec-5/15Seismology
  65. 65. T.K. DattaDepartment Of Civil Engineering, IITMicrozonation using hazard analysisLec-5/16Seismology
  66. 66. T.K. DattaDepartment Of Civil Engineering, IITContd...Lec-5/17Seismology
  67. 67. T.K. DattaDepartment Of Civil Engineering, IITContd...Lec-5/180.35 g0.1 g0.25 g0.4 gDeterministic MicrozonationProbability of exceedance = 0.10.15 g0.4 g0.25 g0.2 g0.1 g0.3 gProbabilistic MicrozonationFig 1.27Seismology
  68. 68. T.K. DattaDepartment Of Civil Engineering, IITLec-1/74
  69. 69. T.K. DattaDepartment Of Civil Engineering, IITChapter -2SEISMIC INPUTS
  70. 70. T.K. DattaDepartment Of Civil Engineering, IITSeismic inputs Various forms of Seismic inputs are used forearthquake analysis of structures. The the form in which the input is provided dependsupon the type of analysis at hand. In addition, some earthquake parameters suchas magnitude, PGA, duration, predominantfrequency etc. may be required. The input data may be provided in time domainor in frequency domain or in both. Further,the input data may be required indeterministic or in probabilistic form. Predictive relationships for different earthquakeparameters are also required in seismic riskanalysis.1/1Seismic Input
  71. 71. T.K. DattaDepartment Of Civil Engineering, IITTime history records The most common way to describe ground motion isby way of time history records. The records may be for displacement, velocityand acceleration; acceleration is generally directlymeasured; others are derived quantities. Raw measured data is not used as inputs; dataprocessing is needed. It includes Removal of noises by filters Baseline correction Removal of instrumental error Conversion from A to D At any measuring station, ground motions arerecorded in 3 orthogonal directions; one is vertical.1/2Seismic Input
  72. 72. T.K. DattaDepartment Of Civil Engineering, IIT They can be transformed to principal directions;major direction is the direction of wave propagation;the other two are accordingly defined. Stochastically, ground motions in principaldirections are uncorrelated.Contd..(a) major (horizontal)Major (horizontal)0 5 10 15 20 25 30 35 40-0.4-0.3-0.2- (sec)Fig 2.1(a)1/3Seismic Input
  73. 73. T.K. DattaDepartment Of Civil Engineering, IITContd..1/40 5 10 15 20 25 30 35 40-0.3-0.2- (sec)Acceleration(g)Minor (horizontal)0 5 10 15 20 25 30 35 40-0.3-0.2- (sec)AcceleratIon(g)Minor (vertical)Fig 2.1(b)Fig 2.1(c)Seismic Input
  74. 74. T.K. DattaDepartment Of Civil Engineering, IIT Because of the complex phenomena involved inthe generation of ground motion, trains of groundmotion recorded at different stations vary spatially. For homogeneous field of ground motion, rms / peakvalues remain the same at two stations but there isa time lag between the two records. For nonhomogeneous field, both time lag & differencein rms exist. Because of the spatial variation of ground motion,both rotational & torsional components of groundmotions are generated.Contd..1/5du dvφ( t ) = + ( 2.1)dy dxdwθ( t ) = ( 2.2)dxSeismic Input
  75. 75. T.K. DattaDepartment Of Civil Engineering, IIT In addition, an angle of incidence of ground motionmay also be defined for the time history record.Contd.. 1/6Major directionxyα =Angle of incidenceFig 2.2Seismic Input
  76. 76. T.K. DattaDepartment Of Civil Engineering, IITFrequency contents of time history Fourier synthesis of time history record providesfrequency contents of ground motion. It provides useful information about the ground motion& also forms the input for frequency domain analysis ofstructure. Fourier series expansion of x(t) can be given as∑∫∫∫a0 n n n nn=1T/20-T/2T/2n n-T/2T/2n n-T/2nx( t ) = a + a cosω t + b sinω t ( 2.3)1a = x( t ) dt ( 2.4)T2a = x( t ) cosω t dt ( 2.5)T2b = x( t ) sinω t dt ( 2.6)Tω = 2πn/T ( 2.7)1/7Seismic Input
  77. 77. T.K. DattaDepartment Of Civil Engineering, IIT The amplitude of the harmonic at is given by(2.8)      ∫∫2T/22 2 2n n n n-T/22T/2n-T/22A = a + b = x( t ) cosω tdtT2+ x( t ) sinω tdtTContd..nω1/8  ÷ n n-1 nnnc = Abφ = tan ( 2.10)a Equation 2.3 can also be represented in the form∑α0 n n nn=1x( t ) = c + c sin(ω t + φ ) ( 2.9)Seismic Input
  78. 78. T.K. DattaDepartment Of Civil Engineering, IIT Plot of cn with is called Fourier Amplitude Spectrum. The integration in Eq. 2.8 is now efficiently performed byFFT algorithm which treats fourier synthesis problem asa pair of fourier integrals in complex domain. Standard input for FFT is N sampled ordinates of timehistory at an interval of ∆t. Output is N complex numbers; first N/2+1 complexquantities provide frequency contents of time historyother half is complex conjugate of the first half.Contd..nω∫∫α-iω t-ααiω t-α1x( iω ) = x( t ) e dt ( 2.11)2πx( t ) = x( iω) e dω ( 2.12)1/9Seismic Input
  79. 79. T.K. DattaDepartment Of Civil Engineering, IITis called Nyquest Frequency. Fourier amplitude spectrum provides agood understanding of the characteristics ofground motion. Spectrums are shown in Fig 2.3. For under standing general nature of spectra, likethose shown in Fig 2.3, spectra of groundaccelerations of many earthquakes areaveraged & smoothed for a particular site.jn2πjω =Tω = Nπ/TContd..( )  ÷ ÷ 1/22 2j j jj-1jjNA = a + b j = 0,....., ( 2.13)2bφ = tan ( 2.14)a1/10Seismic Input
  80. 80. T.K. DattaDepartment Of Civil Engineering, IIT1/11Contd..0 2 4 6 8 10 12 14 16 18 2000. (rad/sec)Fourieramplitude(g-sec)1.4Narrow band0 20 40 60 80 100 120 140 1600123456x 10-3Frequency (rad/sec)OrdinateFourieramplitude(g-sec)Broad bandFig 2.3(a)Fig 2.3(b)Seismic Input
  81. 81. T.K. DattaDepartment Of Civil Engineering, IIT The resulting spectrum plotted on log scale shows: Amplitudes tend to be largest at an intermediaterange of frequency. Bounding frequencies are fc & fmax. fc is inversely proportional to duration. For frequency domain analysis, frequency contentsgiven by FFT provide a better input.Contd.. 1/12Frequency (log scale)fc fmaxOrdinateFourieramplitude(logscale)Fig 2.4Seismic Input
  82. 82. T.K. DattaDepartment Of Civil Engineering, IITExample2.1: 32 sampled values at ∆t = 0.02s aregiven as input to FFT as shown in Fig 2.5YY = 1/16 fft(y,32)9.81nnπω = = 157.07 rad/sT2πdω = = rad/sTContd.. 2/10 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.03-0.02-0.0100. (sec)GroundAcceleration(g)Fig 2.5Seismic Input
  83. 83. T.K. DattaDepartment Of Civil Engineering, IITContd..0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300-0.2- (rad/sec)RealpartAReal part0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300-0.2-0.15-0.1-0.0500.050.10.15Frequency (rad/sec)ImaginarypartAImaginary part2/2Fig 2.6aFig 2.6bSeismic Input
  84. 84. T.K. DattaDepartment Of Civil Engineering, IIT2 2 1/2i i i-1 ii i niA =( a + b ) i = 0....N/2bj = tanω =( 0..dw...w )a Fourier amplitude spectrum is Ai Vs plot & phasespectrum is Φi Vs plot as shown in Fig 2.7Contd.. 2/3iωiωAmplitude spectrum0 20 40 60 80 100 120 140 16000.0050.010.0150.02Frequency (rad/sec)Fourieramplitude(g-sec)Fig 2.7aPhase spectrum0 20 40 60 80 100 120 140 160-1.5-1-0.500.511.5Frequency (rad/sec)Phase(rad)Fig 2.7bSeismic Input
  85. 85. T.K. DattaDepartment Of Civil Engineering, IITPower spectral density function Power spectral density function (PSDF) of groundmotion is a popular seismic input for probabilisticseismic analysis of structures. It is defined as the distribution of the expected meansquare value of the ground motion with frequency. Expected value is a common way of describingprobabilistically a ground motion parameter & isconnected to a stochastic process. The characteristics of a stochastic process is describedlater in chapter 4; one type of stochastic process iscalled ergodic process. For an ergodic process, a single time history of theensemble represents the ensemble characteristics ;ensemble r.m.s is equal to that of the time history.2/4Seismic Input
  86. 86. T.K. DattaDepartment Of Civil Engineering, IIT If future earthquake is assumed as an ergodicprocess, then PSDF of future ground motion (sayacceleration) may be derived using the concept offourier synthesis. Meansquare value of an acceleration time history a(t)using Parsaval’s theorem. PSDF of a(t) is defined as Hence,Contd..∑N/22n01λ = c ( 2.16)2∑∫nω N/2nn=00λ = S( ω ) dω = g( ω ) ( 2.17)2nncS(ω ) = & g( ω ) = S( ω ) dω ( 2.18 )2dω2/5Seismic Input
  87. 87. T.K. DattaDepartment Of Civil Engineering, IIT A close relationship between PSDF & Fourieramplitude spectrum is evident from Eqn. 2.18. A typical PSDF of ground acceleration is shownin Fig 2.8.Contd.. 2/60 10 20 30 40 50 60 7000.511.52Frequency (rad/sec)NormalizedPSDFordinateFig 2.8Seismic Input
  88. 88. T.K. DattaDepartment Of Civil Engineering, IIT Some of the important ground motion parametersare described using the moments of PSDF. Ω is called central frequency denoting concentrationof frequencies of the PSDF. The mean peak accln.(PGA) is defined using Ω, λ, Td. Predominant frequency / period is where PSDF /Fourier spectrum peaks.ω∫nωnn020λ = ω S( ω ) d ( 2.19a)λΩ = ( 2.19b)λ  ÷ && dgmax 02.8ΩTu = 2λ ln ( 2.19c)2πContd.. 2/7Seismic Input
  89. 89. T.K. DattaDepartment Of Civil Engineering, IIT An additional input is needed for probabilistic dynamicanalysis of spatially long structures that have multisupport excitations. The time lag or lack of correlation between excitations atdifferent supports is represented by a coherencefunction & a cross PSDF. The cross PSDF between two excitations which isneeded for the analysis of such structures is given byContd..1 2 1 21 2 1 21 12 2x x x x 1 21 12 2x x x x 1 2 x 1 2S = S S coh( x ,x ,ω ) ( 2.20)S = S S coh( x ,x ,ω ) = S coh( x ,x ,ω) ( 2.21)2/8 More discussions on cross PSDF is given laterin chapter 4.Seismic Input
  90. 90. T.K. DattaDepartment Of Civil Engineering, IIT Records of actual strongmotion records show thatmean square value of theprocess is not stationarybut evolutionary.Contd..2S(ω,t ) = q( t ) S( ω ) ( 2.22)2/9Time(sec)σacc(m/sec2) The earthquake process is better modeled asuniformly modulated stationary process in whichPSDF varies with time as: From the collection of records ,various predictiverelation- ships for cross PSDF, Fourier spectrum,modulating functions have been derived; they aregiven later.Fig 2.9Seismic Input
  91. 91. T.K. DattaDepartment Of Civil Engineering, IITExample2.2: For the time history of Example 2.1, findPSDF.Solution: Using Eqns 2.9, 2.16, 2.18 ordinates of PSDFare obtained. Raw and smoothed PSDFs are shown inFigs 2.10 & 2.11Contd..0 20 40 60 80 100 120 140 16001234x 10-6Frequency (rad/sec)PSDF(g2sec/rad)Fig 2.102/10Seismic Input
  92. 92. T.K. DattaDepartment Of Civil Engineering, IIT Sum of areas of bar = 0.011 (m/s2)2 Area under smoothed PSDF = 0.0113 (m/s2)2 Meansquare value of time history = 0.0112 (m/s2)20 50 100 15000.511.522.53x 10-6Frequency (rad/sec)PSDF(g2sec/rad)Three point averaging(curve fit)Three point averagingFive point averagingFive point averaging(curve fit)Contd..Fig 2.112/11Seismic Input
  93. 93. T.K. DattaDepartment Of Civil Engineering, IIT Response spectrum of earthquake is the mostfavored seismic input for earthquake engineers. There are a number of response spectra used todefine ground motion; displacement, pseudovelocity, absolute acceleration & energy. The spectra show the frequency contents of groundmotion but not directly as Fourier spectrum does. Displacement spectrum forms the basis forderiving other spectra. It is defined as the plot of maximum displacement ofan SDOF system to a particular earthquake as afunction of & ξ. Relative displacement of an SDOF for a given isgiven by (3rd chapter):Response spectrumnω&&gx( t )3/1Seismic Input
  94. 94. T.K. DattaDepartment Of Civil Engineering, IIT At the maximum value of displacement, KE = 0 &hence, If this energy were expressed as KE, then anequivalent velocity of the system would beContd..   ∫∫&&&&nnt-ξω( t-τ)g dn 0vm dnt-ξω( t-τ)v g d0 max1x( t ) = - x(τ ) e sinω( t - τ ) dτ ( 2.23)ωSx = S = ( 2.24a)ωS = x(τ ) e sinω( t - τ ) dτ ( 2.24b)2d1E= kS ( 2.25a)2&&2 2eq deq n d1 1mx = kS ( 2.25b)2 2x =ω S ( 2.25c)3/2Seismic Input
  95. 95. T.K. DattaDepartment Of Civil Engineering, IIT Thus, xeq = Sv; this velocity is called pseudo velocity &is different from the actual maximum velocity. Plots of Sd & Sv over the full range of frequency & adamping ratio are displacement & pseudo velocityresponse spectrums. A closely related spectrum called pseudo accelerationspectrum (spectral acceleration) is defined as: Maximum force developed in the spring of the SDOF is Thus, spectral acceleration multiplied by the massprovides the maximum spring force.Contd..2a n dS =ω S ( 2.26)( ) 2s d n d amaxf = kS = mω S = mS ( 2.27)3/3Seismic Input
  96. 96. T.K. DattaDepartment Of Civil Engineering, IITContd.. This observation shows importance of the spectralacceleration. While displacement response spectrum is the plot ofmaximum displacement, plots of pseudo velocity andacceleration are not so. These three response spectra provide directlysome physically meaningful quantities: Displacement – Maximum deformation Pseudo velocity – Peak SE Pseudo acceleration – Peak force Energy response spectrum is the plot ofagainst a full range of frequency for a specifieddamping ratio; it shows the energy cotents of theground motion at different frequencies.max2E( t )m3/4Seismic Input
  97. 97. T.K. DattaDepartment Of Civil Engineering, IIT At any instant of time t, it may be shown that For ξ = 0, it may further easily be shown that Comparing Eqns.(2.8) & (2.30), it is seen that Fourierspectrum & energy spectrum have similar forms. Fourier amplitude spectrum may be viewed as ameasure of the total energy at the end (t = T) of anundamped SDOF.Contd..      &12 2 2n2E( t )= x( t ) + (ω x( t ) ) ( 2.29)m                    ∫ ∫&& &&12 2 2t tg n g d0 02E( t )= x(τ ) cosω τ dτ + x( τ ) sinω τ dτ ( 2.30)m3/5Seismic Input
  98. 98. T.K. DattaDepartment Of Civil Engineering, IITExample2.3: Draw the spectrums for El Centroacceleration for ξ = 0.05Solution: Using Eqns 2.23 - 2.30, the spectrums aredrawn & are shown in Figs. 2.13 – 2.15Tp(Energy) = 0.55 sTp(Fourier) = 0.58 sTp(Acceleration) = 0.51sContd.. 3/60 0.5 1 1.5 2 2.5 300. period (sec)Energyspectrum(g-sec)Fig 2.13Seismic Input
  99. 99. T.K. DattaDepartment Of Civil Engineering, IIT3/7Contd..0 20 40 60 80 100 120 140 16000.0050.010.0150.02Frequency (rad/sec)Fourieramplitude(g-sec)0 0.5 1 1.5 2 2.5 3 3.5 400. period (Sec)Accelerationresponsespectrum(g)Fig 2.15Fig 2.14Seismic Input
  100. 100. T.K. DattaDepartment Of Civil Engineering, IITD-V-A Spectrum All three response spectra are useful in defining thedesign response spectrum discussed later. A combined plot of the three spectra is thusdesirable & can be constructed because of therelationship that exists between them Some limiting conditions should be realised as T →0 & T→ α. The following conditions (physical) help in plottingthe spectrum.d v na v nlogS = logS - logω ( 2.31)logS = logS + logω ( 2.32)&&d gmaxT→∞a gmaxT→0limS =u ( 3.33)limS =u ( 3.34)3/8Seismic Input
  101. 101. T.K. DattaDepartment Of Civil Engineering, IITFig 2.163/9Seismic Input
  102. 102. T.K. DattaDepartment Of Civil Engineering, IITFig 2.173/10Seismic Input
  103. 103. T.K. DattaDepartment Of Civil Engineering, IIT The response spectrum of El Centro earthquake isidealised by a series of straight lines. Straight lines below a & between points b & c areparallel to Sd axis. Those below f & between d & e are parallel to Sa axis. Below ’a’ shows constant ; below ‘f’ showsconstant . Between b & c constant ; between d & econstant . Left of ‘c’ is directly related to maximum acceleration;right of d is directly related to maximum displacement. Intermediate portion cd is directly related to maximumvelocity of ground motion & most sensitive todamping ratio.Contd.. 3/11&&a gS = ud gS = u&&a a gmaxS =α ud d gmaxS =α uSeismic Input
  104. 104. T.K. DattaDepartment Of Civil Engineering, IIT Response spectrum of many earthquakes showsimilar trend when idealised. This observation led to the construction ofdesign response spectrum using straight lineswhich is of greater importance than responsespectrum of an earthquake.Example2.4: Draw the RSP for Park field earthquakefor & compare it with El Centro earthquakeSolution: Using Eqns. 2.23-2.26, the spectra areobtained & drawn in tripartite plot; it is idealized bystraight lines; Fig 2.18 shows Parkfields & El CentroRSPs. Comparison of Ta to Tf between the two isshown in the book.Contd..%5=ξ3/12Seismic Input
  105. 105. T.K. DattaDepartment Of Civil Engineering, IITFig 2.18Table 2.1 Comparison of periods between Parkfieldand El Centro earthquakes3/13(s) (s) (s) (s) (s) (s)Park field 0.041 0.134 0.436 4.120 12.0 32.0El Centro 0.030 0.125 0.349 3.135 10.0 33.0( )a fT T−aTbTcTdTeTfTSeismic Input
  106. 106. T.K. DattaDepartment Of Civil Engineering, IIT Design response spectrum should satisfy somerequirements since it is intended to be used for safedesign of structures (book-2.5.4) Spectrum should be as smooth as possible. Design spectrum should be representative ofspectra of past ground motions. Two response spectra should be considered tocater to variations & design philosophy. It should be normalized with respect to PGA. Cunstruction of Design Spectrum Expected PGA values for design & maximumprobable earthquakes are derived for the region. Peak values of ground velocity & displacementare obtained as:Design RSP3/14Seismic Input
  107. 107. T.K. DattaDepartment Of Civil Engineering, IITc1 = 1.22 to 0.92 m/s c2 = 6 Plot baseline in four way log paper. Obtain bc, de & cd by using c & d points are fixed; so Tc is known. Tb ≈ Tc/4 ; Ta≈ Tc/10; Te≈10 to 15 s; Tf≈ 30 to 35 s Take from ref(4) given in the book. Sa/g may be plotted in ordinary paper.Contd.. 3/15&& &&&&2gmax gmaxgmax 1 gmax 2gmaxu uu = c ; u = cg u&& &a gmax d gmax v gmaxα u ;α u ;α ua d vα , α & αSeismic Input
  108. 108. T.K. DattaDepartment Of Civil Engineering, IITFig 2.193/160.01 0.02 0.05 0.1 0.2 0.3 0.5 0.7 1 2 3 4 5 6 7 10 20 30 50 70 1000.0010.0020.0030.0040.0050.0070. cT dT eT fTDisp.(m)Pseudovelocity(m/sec)2Acc.(m/sec)& mv guα& mgumDguαmgu&&mAguα&& mguPeak ground acceleration,velocity and displacementElastic design spectraTime period (sec)Seismic Input
  109. 109. T.K. DattaDepartment Of Civil Engineering, IITFig 2.203/170 0.5 1 1.5 2 2.5 3 3.5 400.511.522.53Time period (sec)Sa/gHard soilMedium soilSoft soilTime Period (sec)Pseudo-acceleration(g)Design spectrum for siteMedium-sized earthquake at smallepicentral distanceLarge size earthquake at large epicentral distanceFig 2.21Seismic Input
  110. 110. T.K. DattaDepartment Of Civil Engineering, IITExample2.5: Construct design spectra for the 50thpercentile & 84.1 percentile in Tripartite plot.Solution: Ta = 1/33s; Tb = 1/8s; Te = 10s; Tf = 33sαA, = 2.17(2.71) ; αV = 1.65(2.30)αD =1.39(2.01)For 5 % damping;Values within bracket are for 84.1 percentilespectrum.Plots are shown in Fig 2.22.Contd..& && -1g g2g1.22u = u = 0.732 msg( 0.732)u = = 0.546m0.6g&&gu = 0.6g3/18Seismic Input
  111. 111. T.K. DattaDepartment Of Civil Engineering, IIT3/19Contd..50th84thFig 2.22Seismic Input
  112. 112. T.K. DattaDepartment Of Civil Engineering, IIT Design Earthquake; many different descriptions ofthe level of severity of ground motions are available.Contd.. MCE – Largest earthquake from a source SSE – Used for NP design Other terms denoting similar levels ofearthquake are, credible, safety levelmaximum etc & are upper limits for twolevel concept. Lower level is called as OBE; otherterminologies are operating level,probable design & strength level. OBE ≈ ½ SSE3/20Seismic Input
  113. 113. T.K. DattaDepartment Of Civil Engineering, IIT Site specific spectra are exclusively used for thedesign of structures for the site. It is constructed using recorded earthquake data in& around the site. If needed, earthquake data is augmented byearthquake records of similar geological &geographical regions. Earthquake records are scaled for uniformity &then modified for local soil condition. Averaged & smoothed response spectra obtainedfrom the records are used as site specific spectra.( book – & Example 2.6). The effect of appropriate soil condition may have tobe incorporated by de-convolution and convolutionas shown in Fig 2.23.Site specific spectra4/1Seismic Input
  114. 114. T.K. DattaDepartment Of Civil Engineering, IITContd.. 4/2Fig 2.23Rock outcroping motionCCSoil profile atsite of interestconvolutionESurface motion atsite of interestSurface motionDeconvolutionGiven soilprofileBbedrock motionADBedrock motionsame as point BSeismic Input
  115. 115. T.K. DattaDepartment Of Civil Engineering, IIT Statiscal analysis of available spectrum is performedto find distributions of PGA & spectral ordinate ateach period. From these distributions, values of spectralordinates with specified probability of exceedanceare used to construct the uniform hazard spectra. Alternatively, seismic hazard analysis is carriedout with spectral ordinate (at each period for a givenξ) as parameter (not PGA). From these hazard curves, uniform hazard spectrumfor a given probability of exceedance can beconstructed. An example problem is solved in thebook in order to illustrate the concept. These curvesare used for probabilistic design of structures (book- Example 2.7).Uniform hazard spectra4/3Seismic Input
  116. 116. T.K. DattaDepartment Of Civil Engineering, IIT For many cases, response spectrum or PSDFcompatible time history records are required asinputs for analysis. One such case is nonlinear analysis of structuresfor future earthquakes. Response spectrum compatible ground motionis generated by iteration to match a specifiedspectrum; iteration starts by generating a setof Gaussian random numbers. Many standard programs are now available toobtain response spectrum compatible time histories;brief steps are given in the book (2.6.1). Generation of time history for a given PSDFessentially follows Monte Carlo simulation.Synthetic accelerograms4/4Seismic Input
  117. 117. T.K. DattaDepartment Of Civil Engineering, IIT By considering the time history as a summationof sinusoids having random phase differences,the time history is generated. Relationship between discussedbefore is used to find amplitudes of thesinusoids (book – 2.6.2). Random phase angle, uniformly distributedbetween , is used to find Generation of partially correlated groundmotions at a number of points having the samePSDF is somewhat involved & is given in ref(6) & Sdω4/50 -2π iφ∑ i i iia( t ) = A sin(ω t + φ ) ( 2.39)Seismic Input
  118. 118. T.K. DattaDepartment Of Civil Engineering, IIT Many seismic input parameters & ground motionparameters are directly available from recordeddata; many are obtained using empiricalrelationships. These empirical relationships are not only usedfor predicting future earthquake parameters but alsoare extensively used where scanty data areavailable. Predictive relationships generally express theseismic parameters as a function of M, R, Si ( orany other parameter). They are developed based on certain considerations.Prediction of seismic input parameters( )iY = f M, R, S ( 2.40) The parameters are approximately lognormally distributed.4/6Seismic Input
  119. 119. T.K. DattaDepartment Of Civil Engineering, IIT Decrease in wave amplitude with distance bearsan inverse relationship. Energy absorption due to material dampingcauses amplitudes to decrease exponentially. Effective epicentral distance is greater than R. The mean value of the parameter is obtainedfrom the predictive relationship; a standarddeviation is specified. Probability of exceedance is given by: p is defined byContd..[ ] ( )1P Y≥ Y = 1 - F p ( 2.41)( )1lnYlnY - lnYp = ( 2.42)σ4/7Seismic Input
  120. 120. T.K. DattaDepartment Of Civil Engineering, IIT lnY is the mean value ( in ln ) of the parameter. Many predictive relationships, laws &empirical equations exist; most widely used onesare given in the book. Predictive relationships for different seismicparameters given in the book include.Contd.. Predictive relationships for PGA , PHA & PHV.(Eqns: 2.43 – 2.57). Predictive relationships for duration (Eqn 2.58). Predictive relationships for arms(Eqns2.59 –2.62) Predictive relationship for Fourier & responsespectra (Eqns 2.63 – 2.68).4/8Seismic Input
  121. 121. T.K. DattaDepartment Of Civil Engineering, IITContd.. Predictive relationships for PSDF (Eqns: 2.69 –2.80). Predictive relationships for modulatingfunction (Eqn 2.22) given in Eqns 2.81 – 2.89and Figs. 2.47 – 2.50 Predictive relationships for coherencefunction (Eqns 2.90– 2.99).Example 2.8:Compare between the values of PHA & PHVcalculated by different empirical equationsfor M=7; r=75 & 120 km .Note that PHA denotes generallypeak ground acceleration and PHV refers to peak groundVelocity.4/9Seismic Input
  122. 122. T.K. DattaDepartment Of Civil Engineering, IIT4/10Contd..Empirical Relationship PHA(g)75 km 120 kmEsteva (Equation 2.43) 0.034 0.015Cambell (Equation 2.44) 0.056 0.035Bozorgina(Equation 2.45) 0.030 0.015Toro(Equation 2.46) 0.072 0.037Trifunac(Equation 2.54) 0.198 0.088Empirical RelationshipPHV(cm/s)75 km 120 kmEsteva (Equation 2.49) 8.535 4.161Joyner (Equation 2.56) 4.785 2.285Rosenblueth (Equation 2.50) 2.021 1.715Table 2.3: Comparison of PHAs obtained by different empirical equations for M=7Table 2.4: Comparison of PHVs obtained by different empirical equations for M=7Seismic Input
  123. 123. T.K. DattaDepartment Of Civil Engineering, IITExample 2.9: Compare between the smoothednormalized Fourier spectrum obtained from El Centroearthquake & that given by McGuire et al. (Eqn 2.68)Solution: Assume and; comparisonis shown in Fig 2.45.Contd.. 4/11HzfHzfc 10;2.0 max ==kmRMmsV ws 100;7;1500 1=== −0M7=wMis calculatedusing Eqn 1.11 as35.4 is selectedso that it matches ElCentro earthquake. 10-210-110010110-1100101102103Frequency (Hz)Fourieramplitude(cm/sec)ElcentroEquation (2.60)Fig 2.45Seismic Input
  124. 124. T.K. DattaDepartment Of Civil Engineering, IITExample 2.10: Compare between normalizedspectrums obtained by IBC, Euro-8, IS 1893 and thatgiven by Boore et al. (Eq.2.66) for M=7; R=50 km &Vs = 400 m/s.Solution: Values of b1, to b6 are taken fromTable3.9(book); Gc = 0; PGA=0.35g (obtained)Comparison is shown in Fig 2.464/12Contd..Fig 2.460 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.511.522.5Time period (sec)Sa/gBooreIS CodeEuro CodeIBC CodeSeismic Input
  125. 125. T.K. DattaDepartment Of Civil Engineering, IITContd..Example 2.11: Compare between the shapes of PSDFs ofground acceleration given by Housner & Jennings (Eqn.2.70); Newmark & Resenbleuth (Eqn 2.71); Kanai andTazimi(Eqns 2.72-2.73) & Clough & Penziene (Eqns 2.74-2.75)Solution: All constant multipliers are removed from theequations to compare the shapes; comparisonis shown in Fig 2.47.4/130 10 20 30 40 50 60 7000.511.52Frequency (rad/sec)NormalizedPSDFofaccelerationHousner and JenningsNewmark and RousenbluethKanai TazimiClough and PenzienFig 2.47Seismic Input
  126. 126. T.K. DattaDepartment Of Civil Engineering, IIT
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