Idealized Single Degree of Freedom Structure Ft Mass

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) Spring Force, kips SLOPE = k = 50 kip/in Damping Force, Kips SLOPE = c = 0. 254 kip-sec/in Inertial Force, kips SLOPE = m = 0. 130 kip-sec 2/in

Equation of Dynamic Equilibrium

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

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

Development of Effective Earthquake Force Ground Motion Time History

RELATIVE TOTAL M M Somewhat Meaningless Total Base Shear

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

Undamped Free Vibration (2) T = 0. 5 seconds 1. 0 Circular Frequency (radians/sec) Cyclic Frequency (cycles/sec, Hertz) Period of Vibration (seconds/cycle)

Periods of Vibration of Common Structures 20 story moment resisting frame 1 story moment resisting frame T=2. 2 sec. T=1. 4 sec. T=0. 2 sec 20 story braced frame 1 story braced frame T=1. 6 sec T=0. 9 sec T=0. 1 sec

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

Damped Free Vibration (3)

Undamped Harmonic Loading Equation of Motion: = Frequency of the forcing function = 0. 25 Seconds po=100 kips

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

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

Undamped Resonant Response Curve Linear Envelope

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

Response Ratio: Steady State to Static (Absolute Values) Resonance Slowly Loaded 1. 00 Rapidly Loaded

Damped Harmonic Loading Equation of Motion: po=100 kips

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

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

Damped Harmonic Loading (5% Damping)

Resonance Slowly Loaded Rapidly Loaded

Alternative Form of the Equation of Motion: Divide by m: but Therefore: or and

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

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

Development of an Elastic Response Spectrum

Spectral Response Acceleration, Sa NEHRP Recommended Provisions Use a Smoothed Design Acceleration Spectrum “Short Period” Acceleration 2 SDS SD 1 1 T 0 1 3 TS Period, T “Long Period” Acceleration 2 3 T = 1. 0

Average Acceleration Spectra for Different Site Conditions