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. 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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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

SLOPE = c

= 0.254 kip-sec/in

SLOPE = m

= 0.130 kip-sec2/in


Equation of Dynamic Equilibrium


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


RELATIVE

TOTAL

M

M

Somewhat Meaningless

Total Base Shear


Equation of Motion:

Undamped Free Vibration

Initial Conditions:

Assume:

Solution:


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 frameT=2.2 sec.

10 story moment resisting frameT=1.4 sec.

1 story moment resisting frameT=0.2 sec

20 story braced frameT=1.6 sec

10 story braced frameT=0.9 sec

1 story braced frameT=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:

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


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

Rapidly

Loaded

1.00


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:

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 theEquation of Motion

Equation of Motion:

Divide by m:

but

and

or

Therefore:


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


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


Average Acceleration Spectra for Different Site Conditions


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