structural dynamics of linear elastic single-degree...

SDOF Dynamics 3 - 1Instructional Material Complementing FEMA 451, Design Examples

Structural Dynamics ofLinear Elastic Single-Degree-of-Freedom

(SDOF) Systems


Structural Dynamics

• Equations of motion for SDOF structures

• Structural frequency and period of vibration

• Behavior under dynamic load

• Dynamic magnification and resonance

• Effect of damping on behavior

• Linear elastic response spectra


Importance in Relation to ASCE 7-05• Ground motion maps provide ground

accelerations in terms of response spectrumcoordinates.

• Equivalent lateral force procedure gives base shear in terms of design spectrum and period of vibration.

• Response spectrum is based on 5% critical damping in system.

• Modal superposition analysis uses design response spectrum as basic ground motion input.


Idealized SDOF Structure

Mass

Stiffness

Damping

F t u t( ), ( )

t

F(t)

t

u(t)


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( ) ( ) ( ) ( )+ + =

Equation of Dynamic Equilibrium


-40

0

40

0.00 0.20 0.40 0.60 0.80 1.00

-0.50

0.00

0.50

0.00 0.20 0.40 0.60 0.80 1.00

-15.00

0.00

15.00

0.00 0.20 0.40 0.60 0.80 1.00

-400.00

0.00

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, sec


Observed Response of Linear SDOF(Development of Equilibrium Equation)

-30.00

-15.00

0.00

15.00

30.00

-0.60 -0.30 0.00 0.30 0.60

Displacement, inches

-4.00

-2.00

0.00

2.00

4.00

-20.00 -10.00 0.00 10.00 20.00

Velocity, In/sec

-50.00

-25.00

0.00

25.00

50.00

-500 -250 0 250 500


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

Slope = k= 50 kip/in

Slope = c= 0.254 kip-sec/in

Slope = m= 0.130 kip-sec2/in

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


F t( )f tI ( )

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

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

Equation of Dynamic Equilibrium

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


Mass

• Includes all dead weight of structure• May include some live load• Has units of force/acceleration

Inte

rnal

For

ce

Acceleration

1.0M

Properties of Structural Mass


Damping

• In absence of dampers, is called inherent damping• Usually represented by linear viscous dashpot• Has units of force/velocity

Dam

ping

For

ce

Velocity

1.0C

Properties of Structural Damping


Damping

Dam

ping

For

ce

Displacement

Properties of Structural Damping (2)

Damping vs displacement response iselliptical for linear viscous damper.

AREA =ENERGYDISSIPATED


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

Sprin

g Fo

rce

Displacement

1.0K

Properties of Structural StiffnessS

tiffn

ess


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

Sprin

g Fo

rce

Displacement

Properties of Structural Stiffness (2)S

tiffn

ess

AREA =ENERGYDISSIPATED


Undamped Free Vibration

)cos()sin()( 00 tututu ωω

ω+=

&

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

0u&Initial conditions:

ω0uA&

= B u= 0Solution: ω =km

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

0u


ω =km

f =ωπ2

Tf

= =1 2π

ω

Period of Vibration(sec/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

Dis

plac

emen

t, in

ches

T = 0.5 sec

u0

&u01.0


Approximate Periods of Vibration(ASCE 7-05)

xnta hCT =

NTa

1.0=

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 systems

Note: This applies ONLY to building structures!

For moment frames < 12 stories in height, minimumstory height of 10 feet. N = number of stories.


Empirical Data for Determinationof Approximate Period for Steel Moment Frames

8.0028.0 na hT =


Periods of Vibration of Common Structures

20-story moment resisting frame T = 1.9 sec10-story moment resisting frame T = 1.1 sec1-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


SD1 Cu> 0.40g 1.4

0.30g 1.40.20g 1.50.15g 1.6< 0.1g 1.7

computedua TCTT ≤=

Adjustment Factor on Approximate Period(Table 12.8-1 of ASCE 7-05)

Applicable ONLY if Tcomputed comes from a “properlysubstantiated analysis.”


If you do not have a “more accurate” period (from a computer analysis), you must use T = Ta.

If you have a more accurate period from a computeranalysis (call this Tc), then:

if Tc > CuTa use T = CuTa

if Ta < Tc < TuCa use T = Tc

if Tc < Ta use T = Ta

Which Period of Vibration to Usein ELF Analysis?


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 0&Initial conditions:

Solution:

Assume: u t e st( ) =

ξω

= =c

mccc2 ω ω ξD = −1 2


ξω

= =c

mccc2

Damping in Structures

cc is the critical damping constant.

Time, sec

Displacement, in

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

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

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


-30.00

-15.00

0.00

15.00

30.00

-0.60 -0.30 0.00 0.30 0.60


-4.00

-2.00

0.00

2.00

4.00

-20.00 -10.00 0.00 10.00 20.00

Velocity, In/sec

-50.00

-25.00

0.00

25.00

50.00

-500 -250 0 250 500


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

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

Damping in Structures


Damped Free Vibration (2)

-3-2-10123

0.0 0.5 1.0 1.5 2.0

Time, seconds

Dis

plac

emen

t, in

ches

0% Damping10% Damping20% Damping


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


Inherent damping

Added damping

ξ is a structural (material) propertyindependent of mass and stiffness

critical%0.7to5.0=Inherentξ

ξ is a structural property dependent onmass and stiffness anddamping constant C of device

critical%30to10=Addedξ

Damping in Structures (3)

C


ln uu

1

22

21

=−

πξξ

ξπ

≅−u uu

1 2

22

For alldamping values

For very lowdamping values

Measuring Damping from Free Vibration Test

-1

-0.5

0

0.5

1

0.00 0.50 1.00 1.50 2.00 2.50 3.00Time, Seconds

Ampl

itude

u e t0

−ξω

u1

u2 u3


Undamped Harmonic Loading

)sin()()( 0 tptuktum ω=+&&Equation of motion:

-150-100

-500

50100150

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Time, Seconds

Forc

e, K

ips

po=100 kips= 0.25 sec

= frequency of the forcing function

ωπ2

=T

ω

T


Solution:

Particular solution:

Complimentary solution:

u t C t( ) s in ( )= ω

u t A t B t( ) sin( ) cos( )= +ω ω

⎟⎠⎞

⎜⎝⎛ −

−= )sin()sin(

)/(11)( 2

0 ttkptu ω

ωωω

ωω

Undamped Harmonic Loading (2)

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

Assume system is initially at rest:


Define β ωω

=

( )u t pk

t t( ) sin( ) sin( )=−

−02

11 β

ω β ω

Static displacementSteady state

response(at loading frequency)

Transient response(at structure’s frequency)

Loading frequency

Structure’s natural frequency

Undamped Harmonic Loading

Dynamic magnifier


-10-505

10

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

-10-505

10

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

-10-505

10

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Time, seconds

spac

ee

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)

Steady stateresponse (in.)

Transientresponse (in.)

Total response(in.)




Total response(in.)

Loading (kips)

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

-500

-250

0

250

500

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

-150-100

-500

50100150

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

-500

-250

0

250

500

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

-80

-40

0

40

80

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

p,


-80

-40

0

40

80

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

Dis

plac

emen

t, in

.

2π uS

Undamped Resonant Response Curve

Linear envelope




Total response (in.)

Loading (kips)

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

-150-100

-500

50100150

0 .00 0.25 0 .50 0 .75 1 .00 1.25 1 .50 1 .75 2.00

-500

-250

0

250

500

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

-500

-250

0

250

500

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

-80

-40

0

40

80

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

T im e, seconds

p,


Steady state response (in.)

Transient response (in.)

Total response (in.)

Loading (kips)

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

-150-100

-500

50100150

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

-6

-3

0

3

6

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

-6

-3

0

3

6

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

-6

-3

0

3

6

0.00 0.25 0.50 0 .75 1 .00 1 .25 1.50 1.75 2.00

T im e, seconds

p,


-12.00

-8.00

-4.00

0.00

4.00

8.00

12.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Frequency Ratio β

Mag

nific

atio

n Fa

ctor

1/(1

- β2 ) In phase

180 degrees out of phase

Resonance

Response Ratio: Steady State to Static(Signs Retained)


0.00

2.00

4.00

6.00

8.00

10.00

12.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Frequency Ratio β

Mag

nific

atio

n Fa

ctor

1/(1

- β2 )

Response Ratio: Steady State to Static(Absolute Values)

Resonance

Slowlyloaded

1.00

Rapidlyloaded


Damped Harmonic Loading

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

-150-100-50

050

100150

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Time, Seconds

Forc

e, K

ips

po=100 kips

sec25.02==

ωπT


Solution:

Assume system is initially at rest

Particular solution:

Complimentary solution:

u t C t D t( ) sin( ) cos( )= +ω ω

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

Damped Harmonic LoadingEquation of motion:

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

ω ω ξD = −1 2

ξω

=c

m2

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

+ +C t D tsin( ) cos( )ω ω


Transient response at structure’s frequency(eventually damps out)

Steady state response,at loading frequency

D pk

o=−

− +2

1 22 2 2

ξββ ξβ( ) ( )

Damped Harmonic Loading

C pk

o=−

− +1

1 2

2

2 2 2

ββ ξβ( ) ( )

)cos()sin( tDtC ωω +

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


-50

-40

-30

-20

-10

0

10

20

30

40

50

0.00 1.00 2.00 3.00 4.00 5.00

Time, Seconds

Dis

plac

emen

t Am

plitu

de, I

nche

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

Damped Harmonic Loading (5% Damping)


Staticδξ21

-50

-40

-30

-20

-10

0

10

20

30

40

50

0.00 1.00 2.00 3.00 4.00 5.00

Time, Seconds

Dis

plac

emen

t Am

plitu

de, I

nche

sDamped Harmonic Loading (5% Damping)


Harmonic Loading at ResonanceEffects of Damping

-200

-150

-100

-50

0

50

100

150

200

0.00 1.00 2.00 3.00 4.00 5.00

Time, Seconds

Dis

plac

emen

t Am

plitu

de, I

nche

s

0% Damping %5 Damping


0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Frequency Ratio, β

Dyn

amic

Res

pons

e A

mpl

ifier

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 near 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.


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.


LOADING YIELDING

UNLOADING UNLOADED

F F

F

u

F

u

u u

EnergyStored

EnergyDissipated

EnergyRecovered

TotalEnergyDissipated

CONCEPT of ENERGY STOREDand Energy DISSIPATED

12

1

3

2

4

3


Time, T

F(t)

General Dynamic Loading


General Dynamic Loading Solution Techniques

• Fourier transform

• Duhamel integration

• Piecewise exact

• Newmark techniques

All techniques are carried out numerically.


dt

( ) odFF Fdt

τ τ= +

dF

dt

τ

Piecewise Exact Method

Fo


Initial conditions 00, =ou 00, =ou&

Determine “exact” solution for 1st time step

)(1 τuu = )(1 τuu && = )(1 τuu &&&& =

)(1, τuuo = )(1,0 τuu && =Establish new initial conditions

Obtain exact solution for next time step

)(2 τuu = )(2 τuu && = )(2 τuu &&&& =

LOOP




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.


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


Newmark MethodAdvantages:

• Works for inelastic response

Disadvantages:

• Potential numerical error

Note: NONLIN uses the Newmark method forgeneral response history calculations


-0.40

-0.20

0.00

0.20

0.40

0.00 1.00 2.00 3.00 4.00 5.00 6.00

TIME, SECONDS

GR

OU

ND

AC

C, g

Development of Effective Earthquake Force


Earthquake Ground Motion, 1940 El Centro

Many ground motions now are available via the Internet.

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 10 20 30 40 50 60

Time (sec)

Gro

und

Acce

lera

tion

(g's

)

-30

-20

-10

0

10

20

30

40

0 10 20 30 40 50 60

Time (sec)

Gro

und

Velo

city

(cm

/sec

)

-15

-10

-5

0

5

10

15

0 10 20 30 40 50 60

Time (sec)

Gro

und

Disp

lace

men

t (cm

)


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&& ( ) & ( ) ( ) && ( )+ + = −

Development of Effective Earthquake Force

-0.40

-0.20

0.00

0.20

0.40

0.00 1.00 2.00 3.00 4.00 5.00 6.00

TIME, SECONDS

GR

OU

ND

AC

C, g

Ground Acceleration Response History

gu&& tu&&ru&&


)()()()( tumtuktuctum grrr &&&&& −=++

)()()()( tutumktu

mctu grrr &&&&& −=++

ξω2=mc 2ω=

mk

Divide through by m:

Make substitutions:

“Simplified” form of Equation of Motion:

)()()(2)( 2 tutututu grrr &&&&& −=++ ωξωSimplified form:


)()()(2)( 2 tutututu grrr &&&&& −=++ ωξω

Ground motion acceleration history

Structural frequency

Damping ratio

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


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

Response to Ground Motion (1940 El Centro)

-6

-4

-2

0

2

4

6

0 10 20 30 40 50 60

Time (sec)

Stru

ctur

al D

ispl

acem

ent (

in)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 10 20 30 40 50 60

Time (sec)

Gro

und

Acce

lera

tion

(g's)

Excitation applied to structure with given ξ and ω

Peak displacement

Computed response

SOLVER


0

4

8

12

16

0 2 4 6 8 10

PERIOD, Seconds

DIS

PLA

CEM

ENT,

inch

es

The Elastic Displacement Response SpectrumAn elastic displacement response spectrum is a plotof the peak computed relative displacement, ur, for anelastic structure with a constant damping ξ, a varyingfundamental frequency ω (or period T = 2π/ ω), respondingto a given ground motion.

5% damped response spectrum for structureresponding to 1940 El Centro ground motion


-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

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

Time, Seconds

Dis

plac

emen

t, In

ches

ξ = 0.05T = 0.10 secUmax= 0.0543 in.

0.00

2.00

4.00

6.00

8.00

10.00

0.00 0.50 1.00 1.50 2.00

Period, Seconds

Dis

plac

emen

t, In

ches

Computation of Response Spectrum for El Centro Ground Motion

Elastic response spectrum

Computed response


ξ = 0.05T = 0.20 secUmax = 0.254 in.

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

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

Time, Seconds

Dis

plac

emen

t, In

ches

0.00

2.00

4.00

6.00

8.00

10.00

0.00 0.50 1.00 1.50 2.00

Period, Seconds

Dis

plac

emen

t, In

ches

Computation of Response Spectrumfor El Centro Ground Motion


Computed response


ξ = 0.05T = 0.30 secUmax = 0.622 in.

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

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

Time, Seconds

Dis

plac

emen

t, In

ches

0.00

2.00

4.00

6.00

8.00

10.00

0.00 0.50 1.00 1.50 2.00

Period, Seconds

Dis

plac

emen

t, In

ches



Computed response


ξ = 0.05T = 0.40 secUmax = 0.956 in.

-1.20

-0.90

-0.60

-0.30

0.00

0.30

0.60

0.90

1.20

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

Time, Seconds

Dis

plac

emen

t, In

ches

0.00

2.00

4.00

6.00

8.00

10.00

0.00 0.50 1.00 1.50 2.00

Period, Seconds

Dis

plac

emen

t, In

ches



Computed response


ξ = 0.05T = 0.50 secUmax = 2.02 in.

-2.40

-1.80

-1.20

-0.60

0.00

0.60

1.20

1.80

2.40

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

Time, Seconds

Dis

plac

emen

t, In

ches

0.00

2.00

4.00

6.00

8.00

10.00

0.00 0.50 1.00 1.50 2.00

Period, Seconds

Dis

plac

emen

t, In

ches



Computed response


ξ = 0.05T = 0.60 secUmax= -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

Dis

plac

emen

t, In

ches

0.00

2.00

4.00

6.00

8.00

10.00

0.00 0.50 1.00 1.50 2.00

Period, Seconds

Dis

plac

emen

t, In

ches


Computed response


Complete 5% Damped Elastic DisplacementResponse Spectrum for El Centro

Ground Motion

0.00

2.00

4.00

6.00

8.00

10.00

12.00

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

Dis

plac

emen

t, In

ches


Development of PseudovelocityResponse Spectrum

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

0.0 1.0 2.0 3.0 4.0Period, Seconds

Pseu

dove

loci

ty, i

n/se

c

DTPSV ω≡)(

5% damping


0.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

400.0

0.0 1.0 2.0 3.0 4.0Period, Seconds

Pseu

doac

cele

ratio

n, in

/sec

2

DTPSA 2)( ω≡

Development of PseudoaccelerationResponse Spectrum

5% damping


The pseudoacceleration response spectrum represents the total acceleration of the system, not the relative acceleration. It is nearly identical to the true total acceleration response spectrum for lightly damped structures.

Note About the Pseudoacceleration Response Spectrum

0.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

400.0

0.0 1.0 2.0 3.0 4.0Period, Seconds

Pseu

doac

cele

ratio

n, in

/sec

2

5% damping

Peak groundacceleration


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&& ( ) & ( ) ( ) && ( )+ + = −

PSA is TOTAL Acceleration!

-0.40

-0.20

0.00

0.20

0.40

0.00 1.00 2.00 3.00 4.00 5.00 6.00

TIME, SECONDS

GR

OU

ND

AC

C, g

Ground Acceleration Response History

gu&& tu&&ru&&


Difference Between Pseudo-Accelerationand Total Acceleration

(System with 5% Damping)

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

0.1 1 10Period (sec)

Acce

lera

tion

(in/s

ec2 )

Total Acceleration Pseudo-Acceleration


Difference Between Pseudovelocityand Relative Velocity

(System with 5% Damping)

0

5

10

15

20

25

30

35

40

0.1 1 10Period (sec)

Velo

city

(in/

sec)

Relative Velocity Pseudo-Velocity


Displacement Response Spectrafor Different Damping Values

0.00

5.00

10.00

15.00

20.00

25.00

0.0 1.0 2.0 3.0 4.0 5.0Period, Seconds

Dis

plac

emen

t, In

ches

0%5%10%20%

Damping


Pseudoacceleration Response Spectrafor Different Damping Values

0.00

1.00

2.00

3.00

4.00

0.0 1.0 2.0 3.0 4.0 5.0Period, Seconds

Pseu

doac

cele

ratio

n, g

0%5%10%20%

Damping

Peak groundacceleration


Damping Is Effective in Reducing the Response for (Almost) Any Given Period

of Vibration

• An earthquake record can be considered to be the combination of a large number of harmonic components.

• Any SDOF structure will be in near resonance with oneof these harmonic components.

• Damping is most effective at or near resonance.

• Hence, a response spectrum will show reductions due todamping at all period ranges (except T = 0).


Damping Is Effective in Reducing the Response for Any Given Period of

Vibration

Time (sec)

Am

plitu

de

• Example of an artificially generated wave to resemble a real time ground motion accelerogram.

• Generated wave obtained by combining five different harmonic signals, each having equal amplitude of 1.0.

-4.00-2.000.002.004.00

0.0 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60.0 66.0 72.0 78.0 84.0 90.0

`


The Artificial Wave Is the Sum of Five Harmonics

T = 5.0 s

T = 4.0 s

T = 3.0 s

Time (sec)

Am

plitu

de

-1-0.5

00.5

1

0.0 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60.0 66.0 72.0 78.0 84.0 90.0

`

-1-0.5

00.5

1

0.0 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60.0 66.0 72.0 78.0 84.0 90.0

`

-1-0.5

00.5

1

0.0 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60.0 66.0 72.0 78.0 84.0 90.0

`


T = 2.0 s

T = 1.0 s

Time (sec)

Am

plitu

de

Summation

-1-0.5

00.5

1

0.0 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60.0 66.0 72.0 78.0 84.0 90.0

`

-1-0.5

00.5

1

0.0 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60.0 66.0 72.0 78.0 84.0 90.0

`

-4.00-2.000.002.004.00

0.0 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60.0 66.0 72.0 78.0 84.0 90.0

`

The Artificial Wave Is the Sum of Five Harmonics


FFT curve for the combined wave

Frequency (Hz)

Four

ier a

mpl

itude

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Frequency Ratio, β

Dyn

amic

Res

pons

e A

mpl

ifier

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

Damping Reduces the Responseat Each Resonant Frequency


Use of an Elastic Response 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 sec5% critical damping

0.00

2.00

4.00

6.00

8.00

10.00

12.00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Period, Seconds

Dis

plac

emen

t, In

ches

At T = 0.64 sec, displacement = 3.03 in.


Use of an Elastic Response 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 sec5% critical damping

At T = 0.64 sec, pseudoacceleration = 301 in./sec2

0.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

400.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Period, Seconds

Pseu

doac

cele

ratio

n, in

/sec

2

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


Response Spectrum, ADRS Space

0.00

0.20

0.40

0.60

0.80

1.00

0.00 2.00 4.00 6.00 8.00 10.00 12.00


Pseu

doac

cele

ratio

n, g

Diagonal lines representperiod values, T

T = 0.64s


0.1

1

10

100

0.1 1 10 100 1000

Circular Frequency ω, Radiand per Second

PSEU

DO

VELO

CIT

Y, in

/sec

1.0

D=10.0

0.01

.01

0.1

0.1 0.001

1.0

Line of increasingdisplacement

Line of constantdisplacement

Four-Way Log Plot of Response Spectrum

ωPSVD =

Circular Frequency ω(radians/sec)


0.1

1

10

100

0.1 1 10 100 1000


PSEU

DO

VELO

CIT

Y, in

/sec

100

PSA=1000

10000

100

100000

10 1000

10000Line of increasingacceleration

Line of constantacceleration


PSA PSV= ω



0.1

1

10

100

0.1 1 10 100 1000


PSEU

DO

VELO

CIT

Y, in

/sec


10000

1000

100

10

10.1

ACCELERATIO

N, in/se

c2

100

10

1.0

0.1

0.01

0.001

DISPLACEMENT, in



0.10

1.00

10.00

100.00

0.01 0.10 1.00 10.00

PERIOD, Seconds

PSEU

DO

VELO

CIT

Y, in

/sec

1.0

10.0

0.1

0.01

Acceleration, g

0.001

10.0

0.101.0

0.01

0.001

Displac

emen

t, in.

Four-Way Log Plot of Response SpectrumPlotted vs Period


Development of an ElasticResponse Spectrum

0.10

1.00

10.00

100.00

0.01 0.10 1.00 10.00

PERIOD, Seconds

PSEU

DO

VELO

CIT

Y, in

/sec

1.0

10.0

0.1

0.01

Acceleration, g

0.001

10.0

0.10

1.0

0.01

0.001

Displac

emen

t, in.

Problems with Current Spectrum:

It is for a single earthquake; otherearthquakes will have differentCharacteristics.

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


0.1

1

10

100

0.01 0.1 1 10Period, Seconds

Pseu

do V

eloc

ity, I

n/Se

c

0% Damping5% Damping10% Damping20* Damping

1.0

10.0

0.1

0.01

Acceleration, g

0.001

10.0

0.10

1.0

0.01

0.001

Displac

emen

t, in.

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

1940 El Centro, 0.35 g, N-S


0.1

1.0

10.0

100.0

0.01 0.10 1.00 10.00Period, seconds

Pseu

so V

eloc

ity, i

n/se

c

El CentroLoma PrietaNorth RidgeSan FernandoAverage

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

Different earthquakeswill have different spectra.


Smoothed Elastic Response Spectra(Elastic DESIGN Response Spectra)

• Newmark-Hall spectrum

• ASCE 7 spectrum


0.1

1

10

100

0.01 0.1 1 10 100Period (sec)

Disp

lace

men

t (in

)

0% Damping5% Damping10% Damping

Newmark-Hall Elastic Spectrum

Observations

gu&&max

gu&max

gumax

gvv &&&& max→

gvv max→

at short T

at long T

0→v

0→v&&


Very Stiff Structure (T < 0.01 sec)

Total accelerationZero

Ground accelerationRelative displacement


Very Flexible Structure (T > 10 sec)

Relative displacement Total acceleration

Ground displacement Zero


0.1

1

10

100

0.01 0.1 1 10Period, Seconds

Pseu

do V

eloc

ity, I

n/Se

c

0% Damping5% Damping10% Damping20* Damping

1.0

10.0

0.1

0.01

Acceleration, g

0.001

10.0

0.10

1.0

0.01

0.001

Displac

emen

t, in.

1940 El Centro, 0.35 g, N-S

0.35g

12.7 in/s

4.25 in.

Ground Maxima


Damping One Sigma (84.1%) Median (50%)% Critical aa av ad aa av ad

.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

Newmark’s Spectrum Amplification Factorsfor Horizontal Elastic Response


Newmark-Hall Elastic Spectrum

1

3

2 4

65

1) Draw the lines corresponding to max ggg

vvv ,, &&&

2) Draw line from Tb to Tc

Ta Tb Tc Td Te Tf

gAv&&maxα

3) Draw line from Tc to Td

gVv&maxα

4) Draw line from Td to Te

gDvmaxα

5) Draw connecting line from Ta to Tb

6) Draw connecting line from Te to Tf


ASCE 7Uses a Smoothed Design Acceleration Spectrum

“Long period”acceleration

TS T = 1.0Period, T

Spec

tral

Res

pons

e A

ccel

erat

ion,

Sa SDS

SD13

2

1

“Short period”acceleration

1

3

2

S = 0.6 ST

T+0.4 SaDS

0DS

S = Sa DS

S = ST

aD1

Note exceptions at larger periods

TL

4 D1SLa 2TS =T

4


The ASCE 7 Response Spectrum

is a uniform hazard spectrum based onprobabilistic and deterministic seismichazard analysis.

structural dynamics of linear elastic single-degree...

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