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# PSG-Civil 22.03.2014 01

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### PSG-Civil 22.03.2014 01

1. 1. Indian Institute of Technology Delhi (IIT) New Delhi, INDIA Prof. T. K. Datta Department of Civil Engineering, Indian Institute of Technology Delhi Saturday, 22nd, March 2014 IIT Delhi 1
2. 2. IIT Delhi Structural Dynamics for Practicing Civil Engineers 2 Understanding Dynamics and SDOF
3. 3. IIT Delhi Structural Dynamics for Practicing Civil Engineers The excitation is a time-varying force usually expressed as Acceleration time history Pressure time history Force time history Distinction between Static and Dynamic Motions Force is a constant Structure would respond to any “external disturbance” Forc e Buildi ng Respon se Displacement Acceleration Base Shear Inter-storey drift Stresses In static problems… In Dynamic problems… Response is a constant Response is time- varying Response is dependent only on the static load Response is dependent on excitation force, inertial force and dissipative forces 2
4. 4. IIT Delhi Structural Dynamics for Practicing Civil Engineers 4 In static problems… Elastic properties, K Inertial Properties, M Dissipative Properties, C Elastic Properties, K ExcitationForce Time Time Response In Dynamic problems… Buildi ng Buildi ng
5. 5. IIT Delhi Structural Dynamics for Practicing Civil Engineers 5 How do we define the dynamic motion of a building?
6. 6. IIT Delhi Structural Dynamics for Practicing Civil Engineers )(tPKxxCxM t   It all starts with this… Or sometimes with this… )(),( tPxxFxCxM t   6
7. 7. IIT Delhi Structural Dynamics for Practicing Civil Engineers Dynamic Force Equilibrium Equation Let us consider a simple case… )(tPKxxCxM t   gXMtPKxxCxM   )( 7 Let us consider a simple case… Dynamic Force Equilibrium Equation Let us try to understand its each force components…
8. 8. IIT Delhi Structural Dynamics for Practicing Civil Engineers t xMInertial Force 8 Inertial Force Understanding Mass in a better light… Newton’s First Law of Motion All objects have the tendency to resist changes in their state of motion This tendencyis called Inertia maFI   JTI What is inertia?It is the resistance of an object to change its state of motion (magnitude and direction) An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. REST IS A STATE OF MOTION WITH ZERO VELOCITY D’Alembert's Principle Mass as a measureof amountof inertia Direction is opposite to that of motion On Dynamic equilibrium Mass moment of Inertia Inertial Force
9. 9. IIT Delhi Structural Dynamics for Practicing Civil Engineers Idealizatio ns 9 Point of application of Inertial force: At center of mass If there was no concept of inertia force then… Ball would have stopped here! Galileo's EXPERIMENTS No loss of energy due to friction or other means Point Particl e Rigid Body Deformable Idealizatio ns Let’s see Is this sufficient to define the dynamic problem? NOT E
10. 10. IIT Delhi Structural Dynamics for Practicing Civil Engineers Dissipative Force 10 xC Dissipative Force DissipativeForce Velocity n D xF  xcFD  For practical purposes, in the analysis of buildings, a linear relationship maybe assumed, thus This constant of proportionality, is called the damping constant. c Viscous Damping In reality, the dissipative force is a frequency-dependent quantity. It is hard to quantify explicitly different factors for energy loss. Thus an approximate model maybe chosen… Exponentially decaying (for viscous damping) Displacemen t Tim e NOT E
11. 11. IIT Delhi Structural Dynamics for Practicing Civil Engineers 11 Elastic Force Kx Elastic Force This maybe familiar to you from the static analysis… Nevertheless, this is also an integral part of the dynamic force equilibrium Elastic Force Displace ment For a conventional building we will assume it to have a linear relationship. NOT E
12. 12. IIT Delhi Structural Dynamics for Practicing Civil Engineers 12 It is now clear as to why M, C and K are included in the part of your dynamic analysis of a structure Time-varying External force or pressure Time-varying boundary/support condition Recall )( g t XxMxM   Inertial force is the product of inertial mass and “absolute” acceleration Support acceleration How are dynamic forces induced in the structure? A B
13. 13. IIT Delhi Structural Dynamics for Practicing Civil Engineers gXMtPKxxCxM   )( gXMtPKxxCxM   )( gXMtPKxxCxM   )( 13 Earthquake Force Wind Force Blast Force Force induced due to time-varying pressure on building surface. Force induced due to time-varying boundary condition. Force induced due to time-varying blast wave pressure on building surface as well as ground vibration. Earthquake Force Wind Force Blast Force
14. 14. IIT Delhi Structural Dynamics for Practicing Civil Engineers 14 A Schematic diagram for the dynamic force equilibrium equation k1 c1 m1 X Tim e Time period, T Amplitude , A Harmonic Motion t T   2 tAx  sin xAtA dt dx x  cos xAtA dt xd x 22 2 2 sin  NOT E Displacem ent Velocity Accelerati on x A SDOF Spring-Mass-dashpot system SDOF Spring-Mass-dashpot system x
15. 15. IIT Delhi Structural Dynamics for Practicing Civil Engineers 15 tAx  sin tAx  cos tAx  sin2 Earlier we noted that… xMFI xCFD KxFk  We note that, if excitation frequency is increased inertial and dissipative forces increase  DF IF The increase in inertial and dissipative forces due to increase in excitation frequency do not necessarily mean that responses of the building increases. Increase in frequency do signify that inertial forces and dissipative forces can no longer be ignored in the analysis of a building  Problem can no longer be treated as static. Earlier we noted that… Inertial Force Dissipative Force Elastic Force
16. 16. IIT Delhi Structural Dynamics for Practicing Civil Engineers 16 “Engineering judgment is key to structural modelling” We shall now see how the responses of a SDOF system gets affected due to the dynamic characteristics of a building
17. 17. IIT Delhi Structural Dynamics for Practicing Civil Engineers 17 k c m tp sin0 k c m 0p  n D  Equation of motion for a viscous damped SDOF system subjected to harmonic excitation Equation of motion for a viscous damped SDOF system subjected to harmonic excitation Tim e Time period, Amplitude , p0    2 T For ce Harmonic excitation Mass of SDOF system Coefficient of Stiffness Coefficient of Damping Amplitude of excitation force Angular frequency of excitation Natural frequency of SDOF system Damped natural frequency of SDOF system Damping ratio m k n  2 1  nD  nmc 2 tpkuucum  sin0 0gu
18. 18. IIT Delhi Structural Dynamics for Practicing Civil Engineers 18 Damped structure Undamped structure )0(u t e   nT dT Displacem ent Time Effect of damping on free Vibration  tBtAetu DD tn   sincos)( )0(uA  D nuv B    )0()0( where,
19. 19. IIT Delhi Structural Dynamics for Practicing Civil Engineers 19 Amplitu de  2TPeriod, 0)( )( stu tu Total response Steady-state response P (a) Harmonic force; (b) Response of undamped system subjected to harmonic force; ω/ωn = 0.2; u(0)=0; and v(0) = (ωnp0)/k tu v tutu nst n n            sin 1 )0( cos)0()( 2 wt ust sin 1 2   Transi ent Steady- state k p ust 0  n   where,
20. 20. IIT Delhi Structural Dynamics for Practicing Civil Engineers 2 0 DeformationResponse Factor,Rd 1.0 0 180 Phase Angle Frequency Ratio, n 1.0 0 0  0 0 st d u u R  Deformation response factor and phase angle for an undamped system )sin(sin 1 )( 2 twt u tu n st    0)0()0(  vuFor, )sin()(  wtRutu dst n for0 n for180
21. 21. IIT Delhi Structural Dynamics for Practicing Civil Engineers 21 0)( )( stu tu Envelope curve  nT t Response of undamped system to sinusoidal force of frequency ω=ωn; u(0)= v(0)=0
22. 22. IIT Delhi Structural Dynamics for Practicing Civil Engineers 2 2 Transient Steady-State 0)( )( stu tu nT t Total response Steady-state response Response of damped system to harmonic force ω/ωn=0.2, ζ = 0.05 u(0)=0; and v(0) =ωn p0/k tu v tutu nst n n            sin 1 )0( cos)0()( 2 )sin( 4)1( 2222    wt ust Transi ent Steady- state k p ust 0  n   where,           2 1 1 2 tan dampingofbecause)(,0when  tuc nDnntw st t e u tu n     andfor,cos )1(2 )(At resonance
23. 23. IIT Delhi Structural Dynamics for Practicing Civil Engineers 2 3 2 1 2 1 0)( )( stu tu nT t Envelope curve Steady-state amplitudes Response of undamped system with ζ = 0.05 to sinusoidal force of frequency ω=ωn; u(0)= v(0)=0
24. 24. IIT Delhi Structural Dynamics for Practicing Civil Engineers 2 4 Steady state response of damped system (ζ = 0.2 to sinusoidal force for three value of the frequency ratio; (a) = ω/ωn=0.5, (b) ω/ωn=1, (c) ω/ωn=2 0)( )( stu tu 0)( )( stu tu 0)( )( stu tu nT t nT t nT t 29.1dR 5.2dR 32.0dR 5.0 n 0.1 n 2 n
25. 25. IIT Delhi Structural Dynamics for Practicing Civil Engineers 2 5 DeformationResponse Factor,Rd 1.0 0 180Phase Angle Frequency Ratio, n 1.0 0 0  0 0 st d u u R  %01.0 10.0 20.0 70.0 00.1 Deformation response factor and phase angle for a damped system excited by harmonic force
26. 26. IIT Delhi Structural Dynamics for Practicing Civil Engineers 2 6 Solution of SDOF system for sinusoidal excitation consists of two parts: transient and steady state. Transient response depends upon initial conditions u(0) and v(0) and dies down with time for c ≠ 0 ; when c =0, transient response continues forever. If there were no inherent damping in the structure, all structures would have failed due to continuous oscillation (fortunately, this is not so!) Steady state response is of interest for c≠0 Some important Observation from SDOF subjected to Harmonic excitation… Some important Observation from SDOF subjected to Harmonic excitation… 1/ 8
27. 27. IIT Delhi Structural Dynamics for Practicing Civil Engineers Steady state response is sinusoidal like excitation but with a phase lag ϕ. Amplitude of response = static response × DAF; ϕ depends upon ω/ωn and damping. DAF Vs ω/ωn for displacement, velocity and acceleration (Rd, Rv and Ra) reveal many interesting dynamic behaviour of structures. The relation between Ra, Rd, Rv i.e. Rv = (ω/ωn) Rd and Ra = (ω/ωn)2Rd makes it possible to plot them in a single graph in four way logarithmic plot. 2 6 Some important Observation from SDOF subjected to Harmonic excitation… Some important Observation from SDOF subjected to Harmonic excitation… 2/ 8
28. 28. IIT Delhi Structural Dynamics for Practicing Civil Engineers 2 8 Characteristics of Rd Vs ω/ωn () plotMaximum value of Rd takes place not at ω=ωn but at 2 21 n 2.0 DAF  > 1 0.5 to 1.35 ≈ 1 0.5 < 1 1.35 < 0.25 >>2   2 1 dR Some important Observation from SDOF subjected to Harmonic excitation… Some important Observation from SDOF subjected to Harmonic excitation… For rational damping ratio, 3/ 8
29. 29. IIT Delhi Structural Dynamics for Practicing Civil Engineers 2 9 At resonance ф = 900 ;  >2, ф 1800 and  <0.5, ϕ  0. At resonance, damping force predominates and equilibrates the external force. As a thumb rule, frequency of SDOF should be designed such that  should not lie within the bound given by 0.75 ≤  ≤ 1.25; effect of damping is very significant within this range. Effect of damping becomes insignificant for  >1.5 Some important Observation from SDOF subjected to Harmonic excitation… Some important Observation from SDOF subjected to Harmonic excitation… 4/ 8
30. 30. IIT Delhi Structural Dynamics for Practicing Civil Engineers 3 0 Characteristics of Rv Vs Maximum value of Rv takes place at ω=ωn and 0.2  • As a thumb rule, for tow DAF  should not fall within 0.75<<1.25 ; effect of damping is very significant within this range. • Effect of damping becomes insignificant for >1.6.   2 1 maxvR For rational damping ratio, DAF  > 1 0.75 to 1.6 <1 < 0.75 < 1 > 1.6 < 0.35 > 2.5 2.0 Some important Observation from SDOF subjected to Harmonic excitation… Some important Observation from SDOF subjected to Harmonic excitation… 5/ 8
31. 31. IIT Delhi Structural Dynamics for Practicing Civil Engineers 31 Characteristics of Ra Vs  Maximum value of Ra takes place not at ω=ωn, but at • As a thumb rule, for low DAF  should not fall within 0.8<  <1.5 ; effect of damping is very significant within this range. • Effect of damping becomes insignificant for  > 2 For rational value of DAF  > 1 0.75 1 >3 < 1 > 0.75 2.0 2 21    n   2 1 maxaR Some important Observation from SDOF subjected to Harmonic excitation… Some important Observation from SDOF subjected to Harmonic excitation… 6/ 8
32. 32. IIT Delhi Structural Dynamics for Practicing Civil Engineers 3 2 Characteristics of TR Vs  TR denotes the fraction of the vibratory force transmitted to the foundation when an isolator is in between the force and the foundation. For rational damping, 2.0 DAF  > 1 0.5 to 1.4 <1 >2 << 1 > 3 Some important Observation from SDOF subjected to Harmonic excitation… Some important Observation from SDOF subjected to Harmonic excitation… 7/ 8
33. 33. IIT Delhi Structural Dynamics for Practicing Civil Engineers 3 3 For practical design, it is better to avoid the range of  as 0.75<  <1.3 (TR)max is at  =1. TR also denotes the transmission of ground acceleration to the rigid mass attached to a spring dash pot system (idealization of isolator). The same characteristics hold good. Some important Observation from SDOF subjected to Harmonic excitation… Some important Observation from SDOF subjected to Harmonic excitation… 8/ 8
34. 34. IIT Delhi Structural Dynamics for Practicing Civil Engineers 3 4 Modelling of Buildings
35. 35. IIT Delhi Structural Dynamics for Practicing Civil Engineers BED ROCK Primary members resisting seismic forces- Columns (Imposed design consideration) Understanding the deformation profile Assessing the independent dynamics degrees of freedom 35 Modelling of BuildingModelling of Building
36. 36. IIT Delhi Structural Dynamics for Practicing Civil Engineers BED ROCK Primary members resisting seismic forces- Columns (Imposed design consideration) Understanding the deformation profile Assessing the independent dynamics degrees of freedom 36 Modelling of BuildingModelling of Building
37. 37. IIT Delhi Structural Dynamics for Practicing Civil Engineers BED ROCK Primary members resisting seismic forces- Columns (Imposed design consideration) Understanding the deformation profile Assessing the independent dynamics degrees of freedom 3 7 Modelling of Building Modelling of Building
38. 38. IIT Delhi Structural Dynamics for Practicing Civil Engineers Primary members resisting seismic forces- Columns (Imposed design consideration) Understanding the deformation profile Assessing the independent dynamics degrees of freedom 3 8 Modelling of Building Modelling of Building
39. 39. IIT Delhi Structural Dynamics for Practicing Civil Engineers Assessing the independent dynamics degrees of freedom 3 9 In reality a structure will have infinite degrees of freedom. NOT E Modelling of Building Modelling of Building
40. 40. IIT Delhi Structural Dynamics for Practicing Civil Engineers Assessing the independent dynamics degrees of freedom For practical purposes, one degree of freedom is needed to be considered at each floor level.Mass should be attached to dynamic D.O.F. D.O.F. other than dynamic D.O.F. are condensed out. Point mass lumping does not have MI. Floor is assumed to be rigid in its own plane 4 0 Modelling of Building Modelling of Building
41. 41. IIT Delhi Structural Dynamics for Practicing Civil Engineers 3 12 L EI F   3 12 L EI F   2 6 L EI M   2 6 L EI M   For unit lateral displacement, 1 In one single column… 41 Modelling of BuildingModelling of Building
42. 42. IIT Delhi Structural Dynamics for Practicing Civil Engineers k1 k2 m1 m2 1k 2k 1m 2m          211 11 kkk kk K       2 1 0 0 m m M 31 12 L EI k  4 2 Modelling of Building Modelling of Building
43. 43. IIT Delhi Structural Dynamics for Practicing Civil Engineers 1k 2k 1m 2m k1 c1 k2 c2 m1 m2 1c 2c 31 12 L EI k           211 11 kkk kk K       2 1 0 0 m m M 1 2 1 Rayleigh’s Damping KMC  4 3        2 1 x x X Modelling of Building Modelling of Building
44. 44. IIT Delhi Structural Dynamics for Practicing Civil Engineers Understanding the degrees of freedom in a 3- Dimensional model 4 4 Modelling of Building Modelling of Building
45. 45. IIT Delhi Structural Dynamics for Practicing Civil Engineers Understanding the degrees of freedom in a 3- Dimensional model Center of mass 4 5 Modelling of Building Modelling of Building
46. 46. IIT Delhi Structural Dynamics for Practicing Civil Engineers CM The floor slab is assumed to be rigid and the total mass of the floor is lumped at its center of mass. Dynamic D.O.F. are considered at center of mass. The stiffness matrix written in terms of nodal D.O.F. is condensed to the stiffness matrix corresponding to the D.O.F. at center of mass using transfer matrix. Modeling in STAAD.Pro is different. Dynamic D.O.F = 3N (N = No. of Storey) D.O.F at each node CM CM x yθz 4 6 Modelling of Building Modelling of Building
47. 47. IIT Delhi Structural Dynamics for Practicing Civil Engineers When to go for 3D analysis? 2D 47Asymmetri c Symmetric 3D 3D
48. 48. IIT Delhi Structural Dynamics for Practicing Civil Engineers Center of mass Center of rigidity Inertial forces acts at the center of mass. Center of rigidity is the point through which a force, if applied, will produce only a translation motion in that direction. 4 8 Asymmetric Building
49. 49. IIT Delhi Structural Dynamics for Practicing Civil Engineers Coupling between D. O. F at center of mass. 3D analysis is done unless it is torsionally very stiff. For translational component of ground motion, there is torsion.ith floor Shear wall Core wall Center of mass Center of rigidity xe ye 4 9 Asymmetric Building
50. 50. IIT Delhi Structural Dynamics for Practicing Civil Engineers CG of CM lines vertically eccentricity Translation mass and mass moment of inertia are lumped at floor levels on CG of CM lines Asymmetric Building CG of CM lines CMCR CM CR 5 0
51. 51. IIT Delhi Structural Dynamics for Practicing Civil Engineers Center of mass No coupling between D.O.F. at center of mass. Can be easily analyzed with 2D approximation. For translational component of ground motion, no torsion. 51 Symmetric Building
52. 52. IIT Delhi Structural Dynamics for Practicing Civil Engineers Asymmetric Buildings undergo torsion. Symmetric Buildings also undergo torsion because of: • Lack of correlation of wind forces on the face of the wall (Time lag Effect) • Torsional component of ground motion. • Accidental eccentricity Center of mass Symmetric Building Not e 5 2
53. 53. IIT Delhi Structural Dynamics for Practicing Civil Engineers Observations on Asymmetric Building Stiffer sections will carry more load Loads are shared according to the stiffness of elements. Extent of torsion decides the distress of corner columns and edge columns. Corner column is subjected more stresses. 1 2 C1 C2 21 Center of mass Center of rigidity C1 and C2 are columns of same stiffness.  12 C2 is stressed more than C1 53
54. 54. IIT Delhi Structural Dynamics for Practicing Civil Engineers Positioning of the core and shear walls in the building decides the asymmetry and torsional stiffness of the building.  Torsionally stiff but symmetric  Torsionally stiff but Asymmetric 3 5 4 Observations on Asymmetric Building
55. 55. IIT Delhi Structural Dynamics for Practicing Civil Engineers Assumptions The springs are linear. Both yawning and lateral deformation of foundation on the soil is considered. Soil is homogeneous. BED ROCK BED ROCK In soft soil Radiation important Full analysis Substructure analysis with iteration 5 5 Soil-Structure Interaction
56. 56. IIT Delhi Structural Dynamics for Practicing Civil Engineers 5 6 BED ROCK Understanding the motion kh ch kv cv kθ cθ Spring-dashpot model for foundation Soil-Structure Interaction
57. 57. IIT Delhi Structural Dynamics for Practicing Civil Engineers BED ROCK Free-field ground motion Kinematic InteractionInertial Interaction Ground motions maybe amplified at surface with respect to the bed rock. Radiation Soil-Structure Interaction
58. 58. IIT Delhi Structural Dynamics for Practicing Civil Engineers 5 8 Response Analysis
59. 59. IIT Delhi Structural Dynamics for Practicing Civil Engineers 5 9 )(tPKxxCxM t   Coming back to the governing differential equation of motion…        2 1 x x X Coming back to the governing differential equation of motion… How do we now solve this? t X x X “Absolute” acceleration “Relative” velocity w.r.t. support/ground “Relative” displacement w.r.t. support/ground
60. 60. IIT Delhi Structural Dynamics for Practicing Civil Engineers 6 0 How do we now solve this? How do we now solve this? Solving the equation yields us the required response quantity of interest: acceleration, velocity and displacement in the building Basically, we are just solving a differential equation NOT E Modal Analysis (Based on Normal Mode Theory) Numerical Integration Methods METHODS
61. 61. IIT Delhi Structural Dynamics for Practicing Civil Engineers 61 Physical Model Modal Model Physical Space Modal Space k1 c1 k2 c2 m1 m2 1,, 11  222 ,,  Coupled in physical space Decoupled in modal space Equations will need to be solved simultaneously in physical space Equations can be solved individually in modal space and later on be transformed back into physical space in a “simple manner” Modal Analysis (Based on Normal Mode Theory) Modal Analysis (Based on Normal Mode Theory)    M C K
62. 62. IIT Delhi Structural Dynamics for Practicing Civil Engineers 6 2 BED ROCK Physical World Assumptions Building remains elastic during the excitation. One lateral degree-of-freedom at each floor level is considered. Columns are inextensible and weightless. Building is classically damped. 1,, 11  222 ,,  Modal World )(and 2121  For 2 D.O.F. there will be two natural frequencies NOT E Buildi ng Buildi ng
63. 63. IIT Delhi Structural Dynamics for Practicing Civil Engineers 6 3 More on mode shapes… Each natural frequency has an associated mode shape. Free vibration shape of the structure in a natural frequency is called mode shape of the structure. When pulled laterally and allowed to go it vibrates in frequency 1 called fundamental frequency of vibration. In order to make it vibrate in frequency 2, more energy is required. So when allowed to vibrate freely it vibrates in 1It is difficult to make the structure vibrate only in second mode/nth mode. However by resonance test, it is possible to make the structure vibrate almost (nearly) in the nth mode for many structures. For N D.O.F. there will be N natural frequencies and mode shapes More on mode shapes…
64. 64. IIT Delhi Structural Dynamics for Practicing Civil Engineers 6 4 3rd Mode1st Mode 1 2nd Mode 2 3 1st Mode 2nd Mode 3rd Mode Tall chimneys/Frames may have widely Spaced frequencies Suspension bridge may have closely spaced frequencies Mode Shapes- Chimney/Frame/Suspension Bridge Mode Shapes- Chimney/Frame/Suspension Bridge
65. 65. IIT Delhi Structural Dynamics for Practicing Civil Engineers 6 5 1 3 42 Symmetric Symmetric 1st Mode- X dir 3rd Mode- X dir 2nd Mode- Y dir 4th Mode- Y dir X Y Mode Shapes- Asymmetric Building Mode Shapes- Symmetric Building
66. 66. IIT Delhi Structural Dynamics for Practicing Civil Engineers 6 6 X Y θ 1st mode (ω1) Coupled mode Closely spaced frequencies X Y θ 2nd Mode (ω2) X Y Center of mass Center of rigidity Mode Shapes- Asymmetric Building Mode Shapes- Asymmetric Building
67. 67. IIT Delhi Structural Dynamics for Practicing Civil Engineers 6 7 )(tPKxxCxM t            211 11 kkk kk K       2 1 0 0 m m M        2221 1211 cc cc KMC Coupling terms By using Eigen value analysis Eigen vectors (mode shapes), Eigen values (Natural frequencies), These Eigen vectors are then used to transform the physical model to modal model by pre- and post-multiply the equations of motions to yield decoupled equations of motion Let’s find out how to calculate the mode shapes and natural frequencies first… How do we decouple the coupled equations of motion ? How do we decouple the coupled equations of motion ?  2 
68. 68. IIT Delhi Structural Dynamics for Practicing Civil Engineers 6 8 0 KxxMThus, When a structure vibrates freely in any of its modes of vibration, every point of the structure undergoes a SHM with a frequency equal to the natural frequency of that mode. So, it is possible to write tXX  sin0        2 1 x x X        20 10 0 x x X Thus the equation of motion becomes 000 2  KxXM oMXKx 0 oIXKxM  0 1 )( 1 IMM   Pre-multiplying with M-1 we get wher e oXAx 0 2  Classical Eigen Value Problem NOT E Eigen values and Eigen vectors of matrix A give the natural frequencies and mode shapes of the structure. Number (n) of natural frequencies and mode shapes is equal to the size of matrix A. As we stated earlier, free vibration of a structure gives the mode shapes… As we stated earlier, free vibration of a structure gives the mode shapes…
69. 69. IIT Delhi Structural Dynamics for Practicing Civil Engineers 6 9 Mode shapes and natural frequencies are the two key dynamic characteristics of structures. They are used to study and analyse the response of structures to dynamic loads. Mode shapes of structure may be compared with human moods which display intrinsic human characteristics DancingSingin g Compromisi ng Talkativ e Dynami c Aggressive Response to Disturba nce More…More…
70. 70. IIT Delhi Structural Dynamics for Practicing Civil Engineers 7 0 In other words, response of a structure to dynamic force is a weighted summation of its mode shapes. In a similar way, a structure responds to any dynamic disturbance by combining its natural modes of vibration in different proportion. Responds to disturbance by combining different characteristics (moods) in different proportion, say, aggression is having maximum weightage. 1 2 )(1 tq )(2 tq)(tx Weighing functions )()()( 2211 tqtqtx  This theory is called Normal mode theory of dynamics More…More…
71. 71. IIT Delhi Structural Dynamics for Practicing Civil Engineers 71 This theory is called Normal mode theory of Dynamic analysis. Most popular but valid for linear classically damped system. Most attractive feature of the normal mode theory is that is converts the solution of a MDOF system to the summation of the solutions of a number of SDOF systems. This is possible because of the orthogonal property of mode shapes. More…More…        2 1 0 0 m m MT          2 2 2 1 0 0 KT Mode shape, is mass- normalized if NOT E : There are no more coupling terms
72. 72. IIT Delhi Structural Dynamics for Practicing Civil Engineers 7 2  q x x tx })( 2 1                          )(tPqKqCqM TTTT   iiii pqkqcqm   iiii pqkqcqm        Cc T i      Mm T i      Kk T i whe re Each equation represents a SDOF Solution of SDOF to dynamic excitation is straight forward. More…More…     )()({)( 2211 tqtqqtx        21  Back to physical space with…
73. 73. IIT Delhi Structural Dynamics for Practicing Civil Engineers 7 3 For earthquake excitation each SDOF (modal equation) For earthquake excitation each SDOF (modal equation) g T iiii xMIqkqcqm   giiii xqqq   2 2 2 i i i m k   iii mc 2 rir rir i m m 2 factorionparticipatMode    ωi determines dynamic magnification for the mode NOTE Normal mode theory reduces the size of the problem. Normal mode theory gives better insight into the response of the structure through the modal behavior and its contributions. Mode shapes and frequencies are also used for damage detection. Modal Participation for first mode = ~90% (for stiff buildings) NOT E
74. 74. IIT Delhi Structural Dynamics for Practicing Civil Engineers 7 4 Tips for Designers
75. 75. IIT Delhi Structural Dynamics for Practicing Civil Engineers 7 5 Symmetric buildings may under go torsion under earthquake and wind excitation because of • Accidental (uncertain) eccentricity. • Torsional component of ground motion. • Kinematic interaction of foundations with soil. • Lack of Spatial Correlation Salient Points Salient Points
76. 76. IIT Delhi Structural Dynamics for Practicing Civil Engineers 7 6 Asymmetric buildings have generally closely spaced frequencies and coupled modes; pure torsional / pure translational modes are hardly present; more number of modes are required to get good response. For asymmetric buildings, corner and edge colums are stressed more; the degree depends upon the torsional response. Salient Points Salient Points
77. 77. IIT Delhi Structural Dynamics for Practicing Civil Engineers 7 7 Shear walls / core walls relieve column stresses and are beneficial for reducing torsional response. Core walls may undergo significant warping stresses. Drifts are more towards the bottom storey. Higher modes contribute significantly to the bending stress. Salient Points Salient Points
78. 78. IIT Delhi Structural Dynamics for Practicing Civil Engineers 7 8 More number of modes are excited in hard soil as compared to soft soil. Mat foundation tends to alter the free field ground motion leading to somewhat different dynamic behaviour than anticipated. Buildings may undergo considerable rocking motion (in relatively soft soil) and hence, deflection and stresses in members may be more than anticipated. Salient Points Salient Points
79. 79. IIT Delhi Structural Dynamics for Practicing Civil Engineers 7 9 For tall buildings wind induced acceleration at the top storey levels is of great concern. Ductility demand is high near the bottom storey of the buildings. For taller buildings the ductility demand is generally less in the middle storeys compared to the upper and lower storeys. Salient Points Salient Points
80. 80. IIT Delhi Structural Dynamics for Practicing Civil Engineers 8 0 Deviation of storey ductility from the assumed ones increases for taller buildings. Rotation at the joints are actually limited by infill panels, therefore, full ductility may ______________________? Bidirectional interaction effect alters the yielding and ductility of column elements that are generally envisaged; floor acceleration could also be of concern in certain cases. Salient Points Salient Points
81. 81. IIT Delhi Structural Dynamics for Practicing Civil Engineers 8181 Effect of blast is more at lower levels; taller structures have less effects. Behaviour of buildings could be different in soft soil; relatively tall buildings may be more effected in soft soil. Buildings which are irregular in plan have complex dynamic behaviour both due to wind and earthquake. Out of plane failure of brick walls is of more concern in masonry constructions / for infill panels. Salient Points Salient Points
82. 82. IIT Delhi Structural Dynamics for Practicing Civil Engineers 8 2 Thank You