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212 Ketter Hall, North Campus, Buffalo, NY 14260 civil.buffalo

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.

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212 Ketter Hall, North Campus, Buffalo, NY 14260 civil.buffalo

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  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. Multi Degree Of Freedom - MDOF T1 T2 T3 Modal Analysis

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

  4. MDOF Response

  5. Modal Quantities

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

  7. Modal Superposition Total Response can be obtained from all modes by superposition: SRSSn (xi) = ( SUMn (xi2))1/2 VBASE = srssk (Vi) Ffloor J = srssk (FiJ) Approximations in highly damped structures: 1. Use of undamped mode shapes 2. Use of proportional damping

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

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

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

  11. Analysis of Yielding Storied Frame VB

  12. Analysis of Yielding Storied Frame

  13. Case Study Structure

  14. Case Study Structure

  15. Typical Capacity Curve

  16. Typical Capacity Curve

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

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

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

  20. Spectral Evaluation of Response

  21. Equivalent Properties of Linearized System

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

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

  24. Response with Fluid Dampers

  25. Response with Friction Dampers

  26. Response with Wall Dampers

  27. Analysis and Experiments with Fluid Dampers

  28. Analysis and Experiments with Friction Dampers

  29. Analysis and Experiments with Wall Dampers

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

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