1. #1 212 Ketter Hall, North Campus, Buffalo, NY 14260 www.civil.buffalo.edu Fax: 716 645 3733 Tel: 716 645 2114 x 2400 Control of Structural Vibrations Lecture #4 MDOF Structures Analysis Instructor: Andrei M. Reinhorn P.Eng. D.Sc. Professor of Structural Engineering

2. #2 Multi Degree Of Freedom - MDOF T1T1 T2T2 T3T3 Modal Analysis

3. #3 Review Review of MDOF response using ortho-normalized modes

4. #4 MDOF Response

5. #5 Modal Quantities

6. #6

7. #7 Modal Analysis Modal Response Calculations: _______________________________________________________ Spectral Displacement: Spectral Acceleration: _______________________________________________________ Modal Floor Displacement (u or  ) Modal (Floor) Inertia Force Modal Reaction- BASE SHEAR _______________________________________________________

8. #8 Modal Superposition Total Response can be obtained from all modes by superposition: SRSS n (x i ) = ( SUM n (x i 2 )) 1/2 V BASE = srss k (V i ) F floor J = srss k (F iJ ) Approximations in highly damped structures: 1. Use of undamped mode shapes 2. Use of proportional damping

9. #9 Capacity (Curve) of Structure Capacity is a function, not a single value, indicating the strength of the system at a deformed position of the structure

10. #10 Nonlinear Static Procedure  The static procedure is an equivalent instantaneous approximation of MDOF response to determine internal stresses and other …..

11. #11 Analysis of Yielding Storied Frame Nonlinear Static Analysis (Procedure) Perform static analysis with increasing lateral forces, F i, (applied at each floor) incrementally, proportional to the distribution of inertia forces which appear in the dynamic analysis

12. #12 Analysis of Yielding Storied Frame VBVB

13. #13 Analysis of Yielding Storied Frame

14. #14 Case Study Structure

15. #15 Case Study Structure

16. #16 Typical Capacity Curve

17. #17 Typical Capacity Curve

18. #18 Step-by-step DC Procedure [DC = Demand Capacity]

19. #19 Step-by-step DC Procedure [DC = Demand Capacity]

20. #20 “Spectral” Capacity INCLUDING A NUMBER OF MODAL SHAPES:

21. #21 Spectral Evaluation of Response

22. #22 Equivalent Properties of Linearized System

23. #23 Spectral Evaluation - Linearized Procedure Linearized procedures uses equivalent period and damping to evaluate response

24. #24

25. #25 Spectral Evaluation - Inelastic Spectrum Procedure Inelastic Spectrum Procedure uses no approximations but require inelastic spectrum to evaluate response

26. #26 Response with Fluid Dampers

27. #27 Response with Friction Dampers

28. #28 Response with Wall Dampers

29. #29 Analysis and Experiments with Fluid Dampers

30. #30 Analysis and Experiments with Friction Dampers

31. #31 Analysis and Experiments with Wall Dampers

32. #32 Damped Inelastic Spectra Note that it is necessary to adjust the inelastic spectra fro the increased damping: