Idealized Single Degree of Freedom Structure

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Idealized Single Degree of Freedom Structure
F(t)
Mass
t
Damping
Stiffness
u(t)
t
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Equation of Dynamic Equilibrium
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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
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Equation of Dynamic Equilibrium
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Properties of Structural DAMPING (2)
AREA ENERGY DISSIPATED
DAMPING FORCE
DAMPING
DISPLACEMENT
Damping vs Displacement response is Elliptical
for Linear Viscous Damper
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CONCEPT of ENERGY ABSORBED and DISSIPATED
F
ENERGY DISSIPATED
F
ENERGY ABSORBED
u
u
LOADING
YIELDING

ENERGY RECOVERED
ENERGY DISSIPATED
F
F
u
u
UNLOADING
UNLOADED
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Development of Effective Earthquake Force
Ground Motion Time History
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RELATIVE
TOTAL
M
M
Somewhat Meaningless
Total Base Shear
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Undamped Free Vibration
Initial Conditions
Assume
Solution
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Undamped Free Vibration (2)
T 0.5 seconds
1.0
Period of Vibration (seconds/cycle)
Circular Frequency (radians/sec)
Cyclic Frequency (cycles/sec, Hertz)
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Periods of Vibration of Common Structures
20 story moment resisting frame T2.2 sec. 10
story moment resisting frame T1.4 sec. 1 story
moment resisting frame T0.2 sec 20 story braced
frame T1.6 sec 10 story braced frame T0.9
sec 1 story braced frame T0.1 sec
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Damped Free Vibration
Equation of Motion
Initial Conditions
Assume
Solution
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Damped Free Vibration (3)
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Undamped Harmonic Loading
Equation of Motion
Frequency of the forcing function
0.25 Seconds
po100 kips
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Undamped Harmonic Loading
Equation of Motion
Assume system is initially at rest Particular
Solution
Complimentary Solution
Solution
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Undamped Harmonic Loading
LOADING FREQUENCY
Define
Structures NATURAL FREQUENCY
Transient Response (at STRUCTURE Frequency)
Dynamic Magnifier
Steady State Response (At LOADING Frequency)
Static Displacement
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Undamped Resonant Response Curve
Linear Envelope
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Response Ratio Steady State to Static (Signs
Retained)
In Phase
Resonance
180 Degrees Out of Phase
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Response Ratio Steady State to Static (Absolute
Values)
Resonance
Slowly Loaded
Rapidly Loaded
1.00
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Damped Harmonic Loading
Equation of Motion
po100 kips
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Damped Harmonic Loading
Equation of Motion
Assume system is initially at rest Particular
Solution
Complimentary Solution
Solution
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Damped Harmonic Loading
Transient Response, Eventually Damps Out
Solution
Steady State Response
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Damped Harmonic Loading (5 Damping)
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Resonance
Slowly Loaded
Rapidly Loaded
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Alternative Form of theEquation of Motion
Equation of Motion
Divide by m
but
and
or
Therefore
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General Dynamic Loading
For SDOF systems subject to general dynamic
loads, response may be obtained by

• Duhamels Integral
• Time-stepping methods

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Development of an Elastic Displacement Response
Spectrum, 5 Damping
El Centro Earthquake Record
Maximum Displacement Response Spectrum
T0.6 Seconds
T2.0 Seconds
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Development of an Elastic Response Spectrum
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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
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Average Acceleration Spectra for Different Site
Conditions