Earth 2011-lec-06

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Lecture Numbver 6 tharwat sakr

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Earth 2011-lec-06

  1. 1. EARTHQUAKEENGINEERING<br />2.4. MDOF Ground <br /> Excitation<br />2.3.2. Response to General <br /> Dynamic Loading<br />2.4.1. MDOF Equation of Motion<br />2.4.2. MDOF Free Vibrations<br />2.4.3. MDOF Response to <br /> Earthquakes<br />2.4.4. MDOF Modal Analysis<br />
  2. 2. 2.3.2. Forced Vibrations General Loading<br />Common Types of Dynamic Loads<br />Periodic<br />Sinusoidal<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />2<br />
  3. 3. 2.3.2. Forced Vibrations General Loading<br />Common Types of Dynamic Loads<br />Periodic<br />Sinusoidal<br />Other<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />3<br />
  4. 4. 2.3.2. Forced Vibrations General Loading<br />Common Types of Dynamic Loads<br />Periodic<br />Sinusoidal<br />Other<br />Non Periodic<br />Impulse<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />4<br />
  5. 5. 2.3.2. Forced Vibrations General Loading<br />Common Types of Dynamic Loads<br />Periodic<br />Sinusoidal<br />Other<br />Non Periodic<br />Impulse<br />Explosion<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />5<br />
  6. 6. 2.3.2. Forced Vibrations General Loading<br />Common Types of Dynamic Loads<br />Periodic<br />Sinusoidal<br />Other<br />Non Periodic<br />Impulse<br />Explosion<br />Earthquake<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />6<br />
  7. 7. 2.3.2. Forced Vibrations General Loading<br />Response to General Dynamic Loading<br />0 1002.66 <br />0.002 772.421 <br />0.004 582.664 <br />0.006 427.027 <br />0.008 300.089 <br />0.01 197.234 <br />0.012 114.537 <br />0.014 48.6668 <br />0.016 -3.20412 <br />0.018 -43.4678 <br />0.02 -74.1465 <br />0.022 -96.9462 <br />0.024 -113.303 <br />0.026 -124.424 <br />0.028 -131.319 <br />0.03 -134.835 <br />0.032 -135.674 <br />0.034 -134.422 <br />F(t) is given as a relation between time and Force<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />
  8. 8. 2.3.2. Forced Vibrations General Loading<br />Response to General Dynamic Loading<br />The solution is carried out using different numerical Integration techniques as <br />Numerical Evaluation of DuHamel Integral<br />Central Difference Method<br />Wilson -  Method<br />Newmark -  Method<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />8<br />
  9. 9. 2.3.2. Forced Vibrations General Loading<br />Incremental Equation of Motion<br />Subtracting the Equation of Motion at times t and t + t the resulting Incremental equation of motion can be derived as <br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />9<br />
  10. 10. 2.3.2. Forced Vibrations General Loading<br />Newmark -  Method (Linear Acceleration)<br />- Given : m, c, k, xo, vo, ao, Fi<br />- Select Dt<br />- Calculate where<br />- For each step :<br />- Calculate DF where <br />- Calculate where <br />- Calculate Dx where Dx = / <br />- Calculate Dv where<br />- Calculate Da where<br />- Calculate xi+1, vi+1 and ai+1 where<br />xi+1= xi+ Dx, vi+1= vi+ Dv and ai+1 = ai+ Da <br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />
  11. 11. 2.3.2. Forced Vibrations General Loading<br />Response to Impact<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />11<br />
  12. 12. 2.3.2. Forced Vibrations General Loading<br />Response to Impact<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />12<br />
  13. 13. 2.3.3. Response to Ground Acceleration<br />Response to Ground Excitation<br />Equation of Motion<br />Is the Load Equivalent to ground acceleration<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />13<br />
  14. 14. 2.3.3. Response to Ground Acceleration<br />Response to General Ground Excitation<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />14<br />
  15. 15. F1<br />F2<br />F3<br />F4<br />F1<br />F2<br />F3<br />F4<br />2.4.1. MDOF Equation of Motion<br />x4<br />m4<br />x3<br />k4<br />m3<br />k3<br />x2<br />m2<br />k2<br />x1<br />m1<br />k1<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />15<br />15<br />
  16. 16. 2.4.1. MDOF Equation of Motion<br />Mass, Damping, and Stiffness Matrices According to the Number of Degrees of Freedom <br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />16<br />
  17. 17. 2.4.1. MDOF Equation of Motion<br />Acceleration, Velocity, Displacement and Load Vectors According to the Number of Degrees of Freedom<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />17<br />
  18. 18. ..<br />{X} = - w2 {f} (An cos wnt + Bn sin wnt)<br />2.4.2. MDOF Free Vibrations<br />Free Vibrations of MDOF<br />{X} ={f} (An cos wnt + Bn sin wnt)<br />- [M] w2 {f} +[K] {f}= 0<br />Eigen Value problem<br />([K]- [M] w2 ) {f} = 0<br />| [K]- [M] w2 | = 0<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />18<br />
  19. 19. 2.4.3. MDOF Response to Earthquakes<br />Response of MDOF to Ground motion<br />The Same Numerical Techniques are used to determine the response of MDOF Structures to General Dynamic Loads<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />19<br />
  20. 20. 2.4.4. MDOF Mode Superposition<br />Mode Shapes are orthogonal with respect to the mass and stiffness matrices<br /> <> 0 For i=j<br /> = 0 For ij<br /> <> 0 For i=j<br /> = 0 For ij<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />20<br />
  21. 21. 2.4.4. MDOF Mode Superposition<br />Mode Superposition aims at uncoupling of the Equation of Motion (For each DOF)<br />Substitute by<br />Which is the uncoupled Equation of Motion of the MDOF System which can be solved separately for each DOF and combined again<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />21<br />
  22. 22. 2.4.4. MDOF Mode Superposition<br />For Each DOF i<br />Which is the Single Normalized Equation of Motion of DOF Number<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />22<br />
  23. 23. 2.4.4. MDOF Mode Superposition<br />For Earthquake response<br />Is the participation Factor for modal analysis<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />23<br />
  24. 24. 2.4. MDOF Ground Excitation<br />Questions<br /><br />Discuss the free vibration Equation of Motion of Multi DOF Systems and the meaning of Natural periods and Mode shapes<br /><br />What is the Computational merit of mode superposition method<br /><br />Discuss the Meaning of the Participation Factor in Modal Analysis<br />Prof.Dr. OsmanShaalan Earthquake Engineering Dr. TharwatSakr<br />24<br />
  25. 25. 2.3.3. Response to Ground Acceleration<br />Questions<br /><br />Discuss the Techniques of Numerical Integration of The Dynamic Equation of Motion<br /><br />What are the main categories of structures regarding to Damping<br /><br />Use the MATLAB Segment defined to determine the response of the Structure defined in the previous lecture Questions to “Al Aqaba” Earthquake given<br />Prof.Dr. OsmanShaalanEarthquake Engineering Dr. TharwatSakr<br />25<br />

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