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Idealized Single Degree of Freedom Structure

Idealized Single Degree of Freedom Structure. F(t). Mass. t. Damping. Stiffness. u(t). t. Equation of Dynamic Equilibrium. Observed Response of Linear SDOF ( Development of Equilibrium Equation ). Damping Force, Kips. Inertial Force, kips. Spring Force, kips. SLOPE = k = 50 kip/in.

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Idealized Single Degree of Freedom Structure

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  1. Idealized Single Degree of Freedom Structure F(t) Mass t Damping Stiffness u(t) t

  2. Equation of Dynamic Equilibrium

  3. Observed Response of Linear SDOF (Development of Equilibrium Equation) • Damping Force, Kips Inertial Force, kips Spring Force, kips SLOPE = k = 50 kip/in SLOPE = c = 0.254 kip-sec/in SLOPE = m = 0.130 kip-sec2/in

  4. Equation of Dynamic Equilibrium

  5. Properties of Structural DAMPING (2) AREA = ENERGY DISSIPATED DAMPING FORCE DAMPING DISPLACEMENT Damping vs Displacement response is Elliptical for Linear Viscous Damper

  6. CONCEPT of ENERGY ABSORBED and DISSIPATED F ENERGY DISSIPATED ENERGY ABSORBED F u u LOADING YIELDING + ENERGY RECOVERED ENERGY DISSIPATED F F u u UNLOADING UNLOADED

  7. Development of Effective Earthquake Force Ground Motion Time History

  8. RELATIVE TOTAL M M Somewhat Meaningless Total Base Shear

  9. Equation of Motion: Undamped Free Vibration Initial Conditions: Assume: Solution:

  10. Undamped Free Vibration (2) T = 0.5 seconds 1.0 Circular Frequency (radians/sec) Period of Vibration (seconds/cycle) Cyclic Frequency (cycles/sec, Hertz)

  11. Periods of Vibration of Common Structures 20 story moment resisting frame T=2.2 sec. 10 story moment resisting frame T=1.4 sec. 1 story moment resisting frame T=0.2 sec 20 story braced frame T=1.6 sec 10 story braced frame T=0.9 sec 1 story braced frame T=0.1 sec

  12. Damped Free Vibration Equation of Motion: Initial Conditions: Assume: Solution:

  13. Damped Free Vibration (3)

  14. Undamped Harmonic Loading Equation of Motion: = Frequency of the forcing function = 0.25 Seconds po=100 kips

  15. Undamped Harmonic Loading Equation of Motion: Assume system is initially at rest Particular Solution: Complimentary Solution: Solution:

  16. Undamped Harmonic Loading LOADING FREQUENCY Define Structure’s NATURAL FREQUENCY Transient Response (at STRUCTURE Frequency) Dynamic Magnifier Steady State Response (At LOADING Frequency) Static Displacement

  17. Undamped Resonant Response Curve Linear Envelope

  18. Response Ratio: Steady State to Static (Signs Retained) In Phase Resonance 180 Degrees Out of Phase

  19. Response Ratio: Steady State to Static (Absolute Values) Resonance Slowly Loaded Rapidly Loaded 1.00

  20. Damped Harmonic Loading Equation of Motion: po=100 kips

  21. Damped Harmonic Loading Equation of Motion: Assume system is initially at rest Particular Solution: Complimentary Solution: Solution:

  22. Damped Harmonic Loading Transient Response, Eventually Damps Out Solution: Steady State Response

  23. Damped Harmonic Loading (5% Damping)

  24. Resonance Slowly Loaded Rapidly Loaded

  25. Alternative Form of theEquation of Motion Equation of Motion: Divide by m: but and or Therefore:

  26. General Dynamic Loading For SDOF systems subject to general dynamic loads, response may be obtained by: • Duhamel’s Integral • Time-stepping methods

  27. Development of an Elastic Displacement Response Spectrum, 5% Damping El Centro Earthquake Record Maximum Displacement Response Spectrum T=0.6 Seconds T=2.0 Seconds

  28. Development of an Elastic Response Spectrum

  29. 2 2 3 1 3 1 NEHRP Recommended Provisions Use a Smoothed Design Acceleration Spectrum “Short Period” Acceleration SDS “Long Period” Acceleration Spectral Response Acceleration, Sa SD1 T0 TS T = 1.0 Period, T

  30. Average Acceleration Spectra for Different Site Conditions

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