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Idealized Single Degree of Freedom StructurePowerPoint Presentation

Idealized Single Degree of Freedom Structure

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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

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

Equation of Motion:

= Frequency of the forcing function

= 0.25 Seconds

po=100 kips

Equation of Motion:

Assume system is initially at rest

Particular Solution:

Complimentary Solution:

Solution:

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

Response Ratio: Steady State to Static

(Absolute Values)

Resonance

Slowly

Loaded

Rapidly

Loaded

1.00

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

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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