sdof dynamics

Structural Dynamics

• Equations of Motion for SDOF Structures

• Structural Frequency and Period of Vibration

• Behavior under Dynamic Load

• Dynamic Magnification

• Effect of Damping on Behavior

• Linear Elastic Response Spectra

SDOF Dynamics 1Revised 3/09/06

Idealized Single Degree of Freedom Structure

F(t)Mass

StiffnessDamping

F t u t( ), ( )

SDOF Dynamics 2

Equation of Dynamic Equilibrium

F t( )f tI ( )

f tD ( )0 5. ( )f tS0 5. ( )f tS

F t f t f t f tI D S( ) ( ) ( ) ( )− − − = 0

f t f t f t F tI D S( ) ( ) ( ) ( )+ + =

SDOF Dynamics 3

SDOF Dynamics 4

0.00 0.20 0.40 0.60 0.80 1.00

-15.00

0.00 0.20 0.40 0.60 0.80 1.00

-400.00

400.00

0.00 0.20 0.40 0.60 0.80 1.00

Acceleration, in/sec2

Velocity, in/sec

Displacement, in

Applied Force, kips

Observed Response of Linear SDOF

Time, Seconds

Observed Response of Linear SDOF(Development of Equilibrium Equation)

-30.00

-15.00

-0.60 -0.30 0.00 0.30 0.60

Displacement, inches

-20.00 -10.00 0.00 10.00 20.00

Velocity, In/sec

-50.00

-25.00

-500 -250 0 250 500

Spring Force, kips Damping Force, Kips Inertial Force, kips

SLOPE = m= 0.130 kip-sec2/in

SLOPE = k= 50 kip/in

SLOPE = c= 0.254 kip-sec/in

f t m u tI ( ) ( )=f t c u tD ( ) ( )=f t k u tS ( ) ( )=

SDOF Dynamics 5

Equation of Dynamic EquilibriumF t( )f tI ( )

f tD ( )0 5. ( )f tS0 5. ( )f tS

f t f t f t F tI D S( ) ( ) ( ) ( )+ + =

m u t c u t k u t F t( ) ( ) ( ) ( )+ + =

SDOF Dynamics 6

Properties of Structural MASS

• Includes all dead weight of structure• May include some live load• Has units of FORCE/ACCELERATION

ACCELERATION

SDOF Dynamics 7

Properties of Structural DAMPING

DAMPING

• In absence of dampers, is called Natural Damping• Usually represented by linear viscous dashpot• Has units of FORCE/VELOCITY

VELOCITY

SDOF Dynamics 8

DAMPING

DISPLACEMENT

Properties of Structural DAMPING (2)

AREA =ENERGYDISSIPATED

Damping vs Displacement response isElliptical for Linear Viscous Damper

SDOF Dynamics 9

Properties of Structural STIFFNESS

• Includes all structural members• May include some “seismically nonstructural” members• Requires careful mathematical modelling• Has units of FORCE/DISPLACEMENT

DISPLACEMENT

SDOF Dynamics 10

Properties of Structural STIFFNESS (2)

• Is almost always nonlinear in real seismic response• Nonlinearity is implicitly handled by codes• Explicit modelling of nonlinear effects is possible

DISPLACEMENT

AREA =ENERGYDISSIPATED

SDOF Dynamics 11

Undamped Free Vibration

m u t k u t( ) ( )+ = 0Equation of Motion:

u u0 0Initial Conditions:

SDOF Dynamics 12

)cos()sin()( 00 tututu ωω

ω0uA = B u= 0 ω =

u t A t B t( ) sin( ) cos( )= +ω ωAssume:

Solution:

ω =km

f =ωπ2

= =1 2π

Period of Vibration(seconds/cycle)

Cyclic Frequency(cycles/sec, Hertz)

Circular Frequency (radians/sec)

Undamped Free Vibration (2)

-3-2-10123

0.0 0.5 1.0 1.5 2.0

Time, seconds

T = 0.5 seconds

SDOF Dynamics 13

Approximate Periods of Vibration (ASCE 7-05)

xnta hCT =

Ct = 0.028, x = 0.8 for Steel Moment FramesCt = 0.016, x = 0.9 for Concrete Moment FramesCt = 0.030, x = 0.75 for Eccentrically Braced FramesCt = 0.020, x = 0.75 for All Other SystemsNote: Buildings ONLY!

1.0=For Moment Frames < 12 stories in height, minimumstory height of 10 feet. N= Number of Stories.

SDOF Dynamics 14

Empirical Data for Determinationof Approximate Period For Steel Moment Frames

8.0028.0 na hT =

SDOF Dynamics 15

Periods of Vibration of Common Structures

20 story moment resisting frame T = 1.9 sec.10 story moment resisting frame T = 1.1 sec.1 story moment resisting frame T = 0.15 sec

20 story braced frame T = 1.3 sec10 story braced frame T = 0.8 sec1 story braced frame T = 0.1 sec

Gravity Dam T = 0.2 secSuspension Bridge T = 20 sec

SDOF Dynamics 16

Adjustment Factor on Approximate Period(ASCE 7-05)

computedua TCTT ≤=

SD1 Cu> 0.40g 1.4

0.30g 1.40.20g 1.50.15g 1.6< 0.1g 1.7

Applicable ONLY if Tcomputed comes from a “properlysubstantiated analysis”

SDOF Dynamics 17

Damped Free Vibration

u t e u t u u ttD

DD( ) cos( ) sin( )= +

⎣⎢

⎦⎥

−ξω ω ξωω

ω00 0

m u t c u t k u t( ) ( ) ( )+ + = 0Equation of Motion:

u u0 0Initial Conditions:

Solution:

Assume: u t e st( ) =

mccc2 ω ω ξD = −1 2

SDOF Dynamics 18

Damping in Structures

True damping in structures is NOT viscous. However, for lowdamping values, viscous damping allows for linear equations and vastly simplifies the solution.

-30.00

-15.00

-0.60 -0.30 0.00 0.30 0.60

-20.00 -10.00 0.00 10.00 20.00

Velocity, In/sec

-50.00

-25.00

-500 -250 0 250 500

Spring Force, kips Damping Force, Kips Inertial Force, kips

SDOF Dynamics 19

Damped Free Vibration (2)

-3-2-10123

0.0 0.5 1.0 1.5 2.0

Time, seconds

0% Damping10% Damping20% Damping

SDOF Dynamics 20

Damping in Structures

Cc is the Critical Damping Constant

ξ is expressed as a ratio (0.0 < ξ < 1.0) in computations.

Sometimes ξ is expressed as a percent (0 < ξ < 100%)

Displacement, in

Time, sec

Response of Critically Damped System, ξ=1.0 or 100% critical

SDOF Dynamics 21

Damping in Structures (2)Welded Steel Frame ξ = 0.010Bolted Steel Frame ξ = 0.020

Uncracked Prestressed Concrete ξ = 0.015Uncracked Reinforced Concrete ξ = 0.020Cracked Reinforced Concrete ξ = 0.035

Glued Plywood Shear wall ξ = 0.100Nailed Plywood Shear wall ξ = 0.150

Damaged Steel Structure ξ = 0.050Damaged Concrete Structure ξ = 0.075

Structure with Added Damping ξ = 0.250

SDOF Dynamics 22

Damping in Structures (3)Natural Damping

ξ is a structural (material) property,independent of mass and stiffness

ξNATURAL = 05 7 0. . %to critical

Added Dampingξ is a structural property, dependent on

mass and stiffness, anddamping constant C of device

ξADDED = 10 30to critical%

SDOF Dynamics 23

Measuring Damping from Free Vibration TestFOR ALL

DAMPING VALUES

0.00 0.50 1.00 1.50 2.00 2.50 3.00Time, Seconds

u e t0

−ξω

πξξ

FOR VERY LOWDAMPING VALUES

≅−u uu

SDOF Dynamics 24

Undamped Harmonic Loading

)sin()()( 0 tptuktum ω=+Equation of Motion:

-150-100

50100150

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Time, Seconds

po=100 kips= 0.25 Seconds

= Frequency of the Forcing Function

SDOF Dynamics 25

Undamped Harmonic Loading (2)

m u t k u t p t( ) ( ) s in ( )+ = 0 ωEquation of Motion:

Assume system is initially at rest:

Particular Solution: u t C t( ) s in ( )= ω

Complimentary Solution: u t A t B t( ) sin( ) cos( )= +ω ω

Solution:

⎟⎠⎞

⎜⎝⎛ −

−= )sin()sin(

)/(11)( 2

0 ttkptu ω

ωωω

SDOF Dynamics 26

Undamped Harmonic Loading

β ωω

=LOADING FREQUENCY

Define Structure’s NATURAL FREQUENCY

( )u t pk

t t( ) sin( ) sin( )=−

ω β ω

Static DisplacementSteady State

Response(At Loading Frequency)

Transient Response(at Structure’s Frequency)Dynamic Magnifier

SDOF Dynamics 27

SDOF Dynamics 28

-10-505

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

-10-505

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

-10-505

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Time, seconds

t,-200-100

0100200

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

ω π= 2 rad / secω π= 4 rad / sec uS = 5 0. .inβ = 0 5.

LOADING,kips

STEADYSTATERESPONSE, in

TRANSIENTRESPONSE, in.

TOTALRESPONSE, in

SDOF Dynamics 29

STEADYSTATERESPONSE, in

TRANSIENTRESPONSE, in

TOTALRESPONSE, in

LOADING, kips

ω π= 4 rad / secω π≈ 4 rad / sec uS = 5 0. .inβ = 0 99.

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

-150-100

50100150

0.00 0 .25 0 .50 0 .75 1.00 1 .25 1 .50 1 .75 2 .00

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00Time, seconds

SDOF Dynamics 30

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00Time, seconds

2π uS

Undamped Resonant Response Curve

Linear Envelope

SDOF Dynamics 31

STEADYSTATERESPONSE, in.

TOTALRESPONSE, in.

LOADING, kips

ω π= 4 rad / secω π≈ 4 rad / sec uS = 5 0. .inβ = 1 01.

-150-100

50100150

0.00 0 .25 0.50 0 .75 1.00 1.25 1 .50 1.75 2 .00

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

0 .0 0 0 .25 0 .50 0 .75 1 .00 1 .25 1 .50 1 .75 2 .00

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

T im e, seconds

SDOF Dynamics 32

STEADYSTATERESPONSE, in.

TOTALRESPONSE, in.

LOADING, kips.

ω π=8 rad / secω π= 4 rad / sec uS = 5 0. .inβ = 2 0.

-150-100

50100150

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

T im e, seconds

SDOF Dynamics 33

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Frequency Ratio β

- β2 )

Response Ratio: Steady State to Static(Absolute Values)

Resonance

SlowlyLoaded

RapidlyLoaded

Damped Harmonic Loading

m u t cu t k u t p t( ) ( ) ( ) sin( )+ + = 0 ωEquation of Motion:

-150-100-50

100150

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Time, Seconds

po=100 kips

T = =2 0 25πω

. Seconds

SDOF Dynamics 34

Damped Harmonic LoadingEquation of Motion:

m u t cu t k u t p t( ) ( ) ( ) sin( )+ + = 0 ω

Assume system is initially at rest

Particular Solution: u t C t D t( ) sin( ) cos( )= +ω ω

Complimentary Solution:

[ ]u t e A t B ttD D( ) sin( ) cos( )= +−ξω ω ω

m2Solution:ω ω ξD = −1 2

[ ]u t e A t B ttD D( ) sin( ) cos( )= +− ξω ω ω

+ +C t D tsin( ) cos( )ω ω

SDOF Dynamics 35

Damped Harmonic Loading

Transient Response, at Structure’s Frequency(Eventually Damps Out)

+[ ]u t e A t B ttD D( ) sin( ) cos( )= +− ξω ω ω

)cos()sin( tDtC ωω +Steady State Response,at Loading Frequency

− +2

1 22 2 2

ξββ ξβ( ) ( )

− +1

ββ ξβ( ) ( )

SDOF Dynamics 36

Damped Harmonic Loading (5% Damping)

SDOF Dynamics 37

0.00 1.00 2.00 3.00 4.00 5.00

Time, Seconds

sBETA=1 (Resonance)Beta=0.5Beta=2.0

Staticδξ21

0.00 1.00 2.00 3.00 4.00 5.00

Time, Seconds

sDamped Harmonic Loading (5% Damping)

SDOF Dynamics 38

SDOF Dynamics 39

Harmonic Loading at ResonanceEffects of Damping

0.00 1.00 2.00 3.00 4.00 5.00

Time, Seconds

0% Damping %5 Damping

SDOF Dynamics 40

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Frequency Ratio, β

0.0% Damping5.0 % Damping10.0% Damping25.0 % Damping

RD =− +

11 22 2 2( ) ( )β ξβ

Resonance

SlowlyLoaded Rapidly

Loaded

Summary Regarding Viscous Dampingin Harmonically Loaded Systems

• For systems loaded at a frequency less than their natural frequency, the dynamic response exceeds the static response. This is referred to as Dynamic Amplification.

• An undamped system, loaded at resonance, will have an unbounded increase in displacement over time.

SDOF Dynamics 41

Summary Regarding Viscous Dampingin Harmonically Loaded Systems

• Damping is an effective means for dissipating energy in the system. Unlike strain energy, which is recoverable, dissipated energy is not recoverable.

• A damped system, loaded at resonance, will have a limited displacement over time, with the limit being (1/2ξ) times the static displacement.

• Damping is most effective for systems loaded at or near resonance.

SDOF Dynamics 42

Concept of Energy Absorbedand Dissipated

SDOF Dynamics 43

ENERGYDISSIPATED

F FENERGYABSORBED

u uLOADING YIELDING

TOTALENERGYDISSIPATED

ENERGYRECOVEREDF F

u uUNLOADING UNLOADED

General Dynamic Loading

Time, T

SDOF Dynamics 44

General Dynamic Loading Solution Techniques

• Fourier Transform

• Duhamel Integration

• Piecewise Exact

• Newmark Techniques

All techniques are carried out numerically

SDOF Dynamics 45

dtdFF ττ =)(

Piecewise Exact Method

SDOF Dynamics 46

00, =ou00, =ouInitial Conditions

Determine “Exact” Solution for 1st time step

)(1 τuu = )(1 τuu = )(1 τuu =

Establish New Initial Conditions

)(1, τuuo = )(1,0 τuu =

Obtain Exact Solution for next time step

τuu = )(2 τuu = )(2 τuu =

SDOF Dynamics 47

Advantages:

• Exact if load increment is linear• Very computationally efficient

Disadvantages:

• Not generally applicable for inelastic behavior

Note: NONLIN uses the Piecewise Exact Method forResponse Spectrum Calculations

SDOF Dynamics 48

Newmark Techniques

• Proposed by Nathan Newmark• General method that encompasses a family of

different integration schemes• Derived by:

– Development of incremental equations of motion– Assuming acceleration response over short time step

SDOF Dynamics 49

Newmark Techniques

Incremental Equation of Motion:

Solve for CHANGE indisplacement ∆u(t)over some timestep ∆t

)()()()( tFtuktuctum ∆=∆+∆+∆

SDOF Dynamics 50

Newmark TechniquesConstantAverageacceleration

SDOF Dynamics 51

LinearVelocity

QuadraticDisplacement

)(tu∆

Newmark Techniques• Parameters β and γ are chosen by user to arrive at a

numerical approximation• Parameters control

– Stability of technique– Accuracy of technique

• Two common Newmark approximations− γ=1/2 and β=1/4 Constant average acceleration method− γ=1/2 and β=1/6 Linear acceleration method

• Different meanings of acceleration– Newmark techniques refer to structural acceleration– Piecewise exact method refers to ground acceleration

SDOF Dynamics 52

Newmark MethodAdvantages:

• Works for Inelastic Response

Disadvantages:

• Potential Numerical Error

Note: NONLIN uses the Newmark Method forGeneral Time History Calculations

SDOF Dynamics 53

Development of Effective Earthquake Force

SDOF Dynamics 54

0.00 1.00 2.00 3.00 4.00 5.00 6.00

TIME, SECONDS

Earthquake Ground Motion - 1940 El Centro

Many ground motions now available via the Internet

SDOF Dynamics 55

0 10 20 30 40 50 60

Time (sec)

0 10 20 30 40 50 60

Time (sec)

0 10 20 30 40 50 60

Time (sec)

Development of Effective Earthquake Force

0.00 1.00 2.00 3.00 4.00 5.00 6.00

TIME, SECONDS

Ground ACCELERATION Time History

m u t u t c u t k u tg r r r[ ( ) ( )] ( ) ( )+ + + = 0

mu t c u t k u t mu tr r r g( ) ( ) ( ) ( )+ + = −

SDOF Dynamics 56

“Simplified” form of Equation of Motion:

)()()()( tumtuktuctum grrr −=++

)()()()( tutumktu

mctu grrr −=++

Divide through by m:

ξω2=mc 2ω=

Make substitutions:

Simplified form:

)()()(2)( 2 tutututu grrr −=++ ωξω

SDOF Dynamics 57

For a given ground motion, the response history ur(t) is function of the structure’s frequency ω and damping ratio ξ

)()()(2)( 2 tutututu grrr −=++ ωξω

Ground motion acceleration history

Structural frequency

Damping ratio

SDOF Dynamics 58

SDOF Dynamics 59

Change in ground motion or structural parameters ξand ω requires re-calculation of structural response

Response to Ground Motion (1940 El Centro)

0 10 20 30 40 50 60

Time (sec)

0 10 20 30 40 50 60

Time (sec)

Excitation applied to structure with given ξ and ω

Peak Displacement

Computed Response

SOLVER

The Elastic Displacement Response Spectrum

SDOF Dynamics 60

0 2 4 6 8 10

PERIOD, Seconds

is a plot of the peak computed relative displacement(deformation), ur, for an elastic structure with a constantdamping ξ, a varying fundamental frequency ω(or period T=2π/ ω), responding to a given ground motion.

5% Damped Displacement Response Spectrum for StructureResponding to 1940 El Centro Ground Motion

SDOF Dynamics 61

0 1 2 3 4 5 6 7 8 9 10 11 12

Time, Seconds

ξ=0.05T=0.10 SecondsUmax= 0.0543 in.

0.00 0.50 1.00 1.50 2.00

Period, Seconds

Computation of Displacement Response Spectrumfor El Centro Ground Motion

Elastic Response Spectrum

SDOF Dynamics 62

0 1 2 3 4 5 6 7 8 9 10 11 12

Time, Seconds

0.00 0.50 1.00 1.50 2.00

Period, Seconds

SDOF Dynamics 63

0 1 2 3 4 5 6 7 8 9 10 11 12

Time, Seconds

0.00 0.50 1.00 1.50 2.00

Period, Seconds

SDOF Dynamics 64

0 1 2 3 4 5 6 7 8 9 10 11 12

Time, Seconds

0.00 0.50 1.00 1.50 2.00

Period, Seconds

SDOF Dynamics 65

0 1 2 3 4 5 6 7 8 9 10 11 12

Time, Seconds

0.00 0.50 1.00 1.50 2.00

Period, Seconds

SDOF Dynamics 66

ξ=0.05T=0.60 SecondsUmax= -3.00 in.

-3.20-2.40-1.60-0.800.000.801.602.403.20

0 1 2 3 4 5 6 7 8 9 10 11 12

Time, Seconds

0.00 0.50 1.00 1.50 2.00

Period, Seconds

SDOF Dynamics 67

Complete 5% Damped Elastic DisplacementResponse Spectrum for El Centro

Ground Motion

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Period, Seconds

Development of PseudovelocityResponse Spectrum

SDOF Dynamics 68

0.0 1.0 2.0 3.0 4.0Period, Seconds

DTPSV ω≡)(

5% Damping

0.0 1.0 2.0 3.0 4.0Period, Seconds

DTPSA 2)( ω≡

Development of PseudoaccelerationResponse Spectrum

5% Damping

SDOF Dynamics 69

Note about the Pseudoacceleration SpectrumThe Pseudoacceleration Response Spectrum represents theTOTAL ACCELERATION of the system, not the relativeacceleration. It is nearly identical to the true total accelerationresponse spectrum for lightly damped structures.

SDOF Dynamics 70

0.0 1.0 2.0 3.0 4.0Period, Seconds

5% Damping

Difference Between Pseudo-Accelerationand Total Acceleration

(System with 5% Damping)

100.00

150.00

200.00

250.00

300.00

350.00

0.1 1 10Period (sec)

Total Acceleration Pseudo-Acceleration

SDOF Dynamics 71

Displacement Response Spectrafor Different Damping Values

0.0 1.0 2.0 3.0 4.0 5.0Period, Seconds

0%5%10%20%

Damping

SDOF Dynamics 72

Pseudoacceleration Response Spectrafor Different Damping Values

0.0 1.0 2.0 3.0 4.0 5.0Period, Seconds

0%5%10%20%

Damping

SDOF Dynamics 73

Use of an Elastic Displacement SpectrumExample StructureK = 500 k/inW = 2,000 kM = 2000/386.4 = 5.18 k-sec2/inω = (K/M)0.5 =9.82 rad/secT=2π/ω = 0.64 seconds5% Critical Damping

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Period, SecondsD

@T=0.64 seconds, Displacement=3.03 inches

SDOF Dynamics 74

Use of an Elastic Pseudoacceleration SpectrumExample StructureK = 500 k/inW = 2,000 kM = 2000/386.4 = 5.18 k-sec2/inω = (K/M)0.5 =9.82 rad/secT=2π/ω = 0.64 seconds5% Critical Damping

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Period, Seconds

@T=0.64 seconds, Pseudoacceleration = 301 in/sec2

Base Shear = M x PSA = 5.18(301) = 1559 kips

SDOF Dynamics 75

SDOF Dynamics 76

Response Spectrum, ADRS Space

0.00 2.00 4.00 6.00 8.00 10.00 12.00

Diagonal Lines RepresentPeriod Values T

SDOF Dynamics 77

0.1 1 10 100 1000

Circular Frequency ω, Radiand per Second

D=10.0

0.1 0.001

Line of IncreasingDisplacement

Line of ConstantDisplacement

Four-Way Log Plot of Response Spectrum

ωPSVD =

Circular Frequency ω, Radians per Second

0.1 1 10 100 1000

PSA=1000

100000

10 1000

10000Line of IncreasingAcceleration

Line of ConstantAcceleration

SDOF Dynamics 78

PSA PSV= ω

0.1 1 10 100 1000

ACCELERATIO

N, in/se

DISPLACEMENT, in

SDOF Dynamics 79

100.00

0.01 0.10 1.00 10.00

PERIOD, Seconds

Acceleration, g

0.101.0

Displac

t, in.

Four-Way Log Plot of Response SpectrumPlotted vs Period

SDOF Dynamics 80

Development of an ElasticResponse Spectrum

Problems With Current Spectrum:

100.00

0.01 0.10 1.00 10.00

PERIOD, Seconds

Acceleration, g

Displac

t, in.

For a given earthquake,small variations in structural frequency (period) can producesignificantly different results.

It is for a single earthquake. Otherearthquakes will have differentcharacteristics

SDOF Dynamics 81

0.01 0.1 1 10Period, Seconds

ity, I

0% Damping5% Damping10% Damping20* Damping

Acceleration, g

Displac

t, in.

For a given earthquake,small variations in structural frequency (period) can producesignificantly different results.

1940 El Centro, 0.35 g, N-S

SDOF Dynamics 82

5% Damped Spectra for Four California EarthquakesScaled to 0.40 g (PGA)

SDOF Dynamics 83

0.01 0.10 1.00 10.00Period, seconds

ity, i

El CentroLoma PrietaNorth RidgeSan FernandoAverage

Different earthquakeswill have different spectra.

Smoothed Elastic Response Spectrum(Elastic DESIGN Response Spectrum)

• Newmark-Hall Spectrum

• ASCE 7 Spectrum

SDOF Dynamics 84

Newmark-Hall Elastic Spectrum

0.01 0.1 1 10 100Period (sec)

0% Damping5% Damping10% Damping

Observations

gvv max→

at short T

at long T

SDOF Dynamics 85

Very Flexible Structure (T> 10 sec)

SDOF Dynamics 86

Ground Displacement Relative Displacement Total Acceleration Zero

Very Stiff Structure (T< 0.01 sec)

SDOF Dynamics 87

Total AccelerationZero

Ground AccelerationRelative Displacement

SDOF Dynamics 88

0.01 0.1 1 10Period, Seconds

ity, I

0% Damping5% Damping10% Damping20* Damping

Acceleration, g

Displac

t, in.

1940 El Centro, 0.35 g, N-S

12.7 in/s

4.25 in.

Ground Maxima

Newmark’s Spectrum Amplification Factorsfor Horizontal Elastic Response

Damping One Sigma (84.1%) Median (50%)% Critical αa αv αd αa αv αd

.05 5.10 3.84 3.04 3.68 2.59 2.011 4.38 3.38 2.73 3.21 2.31 1.822 3.66 2.92 2.42 2.74 2.03 1.633 3.24 2.64 2.24 2.46 1.86 1.525 2.71 2.30 2.01 2.12 1.65 1.397 2.36 2.08 1.85 1.89 1.51 1.2910 1.99 1.84 1.69 1.64 1.37 1.2020 1.26 1.37 1.38 1.17 1.08 1.01

SDOF Dynamics 89

Newmark-Hall Elastic Spectrum

1) Draw the lines corresponding to max

Ta Tb Tc Td Te

gggvvv ,,

2) Draw line from Tb to Tc

gAvmaxα

3) Draw line from Tc to Td

gVvmaxα

4) Draw line from Td to Te

gDvmaxα

5) Draw connecting line from Ta to Tb

6) Draw connecting line from Te to Tf

SDOF Dynamics 90

ASCE 7 Uses a Smoothed Design Acceleration Spectrum

“Long Period” Acceleration

T0 TS T = 1.0Period, T

“Short Period” Acceleration 1

S = 0.6 ST

T+0.4 SaDS

S = Sa DS

S = ST

T = 0.2 SS

SDOF Dynamics 91

ASCE 7 Response Spectrum

Is a Uniform Hazard Spectrum based onProbabilistic and Deterministic SeismicHazard Analysis.

SDOF Dynamics 92

sdof dynamics

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lec-02 vibration of sdof system

sistem satu derajat kebebasan sdof

ema hun 04 sdof rendszerek áttekintés

contsys1 l6 sdof control