Lecture Numbver 6 tharwat sakr

1. EARTHQUAKEENGINEERING 2.4. MDOF Ground Excitation 2.3.2. Response to General Dynamic Loading 2.4.1. MDOF Equation of Motion 2.4.2. MDOF Free Vibrations 2.4.3. MDOF Response to Earthquakes 2.4.4. MDOF Modal Analysis

2. 2.3.2. Forced Vibrations General Loading Common Types of Dynamic Loads Periodic Sinusoidal Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 2

3. 2.3.2. Forced Vibrations General Loading Common Types of Dynamic Loads Periodic Sinusoidal Other Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 3

4. 2.3.2. Forced Vibrations General Loading Common Types of Dynamic Loads Periodic Sinusoidal Other Non Periodic Impulse Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 4

5. 2.3.2. Forced Vibrations General Loading Common Types of Dynamic Loads Periodic Sinusoidal Other Non Periodic Impulse Explosion Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 5

6. 2.3.2. Forced Vibrations General Loading Common Types of Dynamic Loads Periodic Sinusoidal Other Non Periodic Impulse Explosion Earthquake Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 6

7. 2.3.2. Forced Vibrations General Loading Response to General Dynamic Loading 0 1002.66 0.002 772.421 0.004 582.664 0.006 427.027 0.008 300.089 0.01 197.234 0.012 114.537 0.014 48.6668 0.016 -3.20412 0.018 -43.4678 0.02 -74.1465 0.022 -96.9462 0.024 -113.303 0.026 -124.424 0.028 -131.319 0.03 -134.835 0.032 -135.674 0.034 -134.422 F(t) is given as a relation between time and Force Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr

8. 2.3.2. Forced Vibrations General Loading Response to General Dynamic Loading The solution is carried out using different numerical Integration techniques as Numerical Evaluation of DuHamel Integral Central Difference Method Wilson -  Method Newmark -  Method Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 8

9. 2.3.2. Forced Vibrations General Loading Incremental Equation of Motion Subtracting the Equation of Motion at times t and t + t the resulting Incremental equation of motion can be derived as Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 9

10. 2.3.2. Forced Vibrations General Loading Newmark -  Method (Linear Acceleration) - Given : m, c, k, xo, vo, ao, Fi - Select Dt - Calculate where - For each step : - Calculate DF where - Calculate where - Calculate Dx where Dx = / - Calculate Dv where - Calculate Da where - Calculate xi+1, vi+1 and ai+1 where xi+1= xi+ Dx, vi+1= vi+ Dv and ai+1 = ai+ Da Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr

11. 2.3.2. Forced Vibrations General Loading Response to Impact Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 11

12. 2.3.2. Forced Vibrations General Loading Response to Impact Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 12

13. 2.3.3. Response to Ground Acceleration Response to Ground Excitation Equation of Motion Is the Load Equivalent to ground acceleration Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 13

14. 2.3.3. Response to Ground Acceleration Response to General Ground Excitation Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 14

15. F1 F2 F3 F4 F1 F2 F3 F4 2.4.1. MDOF Equation of Motion x4 m4 x3 k4 m3 k3 x2 m2 k2 x1 m1 k1 Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 15 15

16. 2.4.1. MDOF Equation of Motion Mass, Damping, and Stiffness Matrices According to the Number of Degrees of Freedom Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 16

17. 2.4.1. MDOF Equation of Motion Acceleration, Velocity, Displacement and Load Vectors According to the Number of Degrees of Freedom Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 17

18. .. {X} = - w2 {f} (An cos wnt + Bn sin wnt) 2.4.2. MDOF Free Vibrations Free Vibrations of MDOF {X} ={f} (An cos wnt + Bn sin wnt) - [M] w2 {f} +[K] {f}= 0 Eigen Value problem ([K]- [M] w2 ) {f} = 0 | [K]- [M] w2 | = 0 Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 18

19. 2.4.3. MDOF Response to Earthquakes Response of MDOF to Ground motion The Same Numerical Techniques are used to determine the response of MDOF Structures to General Dynamic Loads Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 19

20. 2.4.4. MDOF Mode Superposition Mode Shapes are orthogonal with respect to the mass and stiffness matrices <> 0 For i=j = 0 For ij <> 0 For i=j = 0 For ij Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 20

21. 2.4.4. MDOF Mode Superposition Mode Superposition aims at uncoupling of the Equation of Motion (For each DOF) Substitute by Which is the uncoupled Equation of Motion of the MDOF System which can be solved separately for each DOF and combined again Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 21

22. 2.4.4. MDOF Mode Superposition For Each DOF i Which is the Single Normalized Equation of Motion of DOF Number Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 22

23. 2.4.4. MDOF Mode Superposition For Earthquake response Is the participation Factor for modal analysis Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 23

24. 2.4. MDOF Ground Excitation Questions  Discuss the free vibration Equation of Motion of Multi DOF Systems and the meaning of Natural periods and Mode shapes  What is the Computational merit of mode superposition method  Discuss the Meaning of the Participation Factor in Modal Analysis Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr 24

25. 2.3.3. Response to Ground Acceleration Questions  Discuss the Techniques of Numerical Integration of The Dynamic Equation of Motion  What are the main categories of structures regarding to Damping  Use the MATLAB Segment defined to determine the response of the Structure defined in the previous lecture Questions to “Al Aqaba” Earthquake given Prof.Dr. OsmanShaalanEarthquake Engineering Dr. TharwatSakr 25