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## PowerPoint Slideshow about 'Idealized Single Degree of Freedom Structure' - waverly

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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

Properties of Structural DAMPING (2)

AREA =

ENERGY

DISSIPATED

DAMPING FORCE

DAMPING

DISPLACEMENT

Damping vs Displacement response is

Elliptical for Linear Viscous Damper

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

Development of Effective Earthquake Force

Ground Motion Time History

Undamped Free Vibration (2)

T = 0.5 seconds

1.0

Circular Frequency

(radians/sec)

Period of Vibration

(seconds/cycle)

Cyclic Frequency

(cycles/sec, Hertz)

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

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:

Solution:

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

Undamped Resonant Response Curve

Linear Envelope

Damped Harmonic Loading

Equation of Motion:

Assume system is initially at rest

Particular Solution:

Complimentary Solution:

Solution:

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

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

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