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

t

F(t)Mass

StiffnessDamping

F t u t( ), ( )

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

-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, Seconds

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 = 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

MASS

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

INE

RTI

AL

FOR

CE

ACCELERATION

1.0M

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

DA

MP

ING

FO

RC

E

VELOCITY

1.0C

SDOF Dynamics 8

DAMPING

DA

MP

ING

FO

RC

E

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

SPR

ING

FO

RC

E

DISPLACEMENT

1.0K

STI

FFN

ES

S

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

SPR

ING

FO

RC

E

DISPLACEMENT

STI

FFN

ES

S

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

ω+=

ω0uA = B u= 0 ω =

km

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

Solution:

ω =km

f =ωπ2

Tf

= =1 2π

ω

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

Dis

plac

emen

t, in

ches

T = 0.5 seconds

u0

u01.0

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!

NTa

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( ) =

ξω

= =c

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

0.00

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

SDOF Dynamics 19

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

SDOF Dynamics 20

ξω

= =c

mccc2

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

C

ξADDED = 10 30to critical%

SDOF Dynamics 23

Measuring Damping from Free Vibration TestFOR ALL

DAMPING VALUES

-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

ln uu

1

22

21

=−

πξξ

FOR VERY LOWDAMPING VALUES

ξπ

≅−u uu

1 2

22

SDOF Dynamics 24

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 Seconds

= Frequency of the Forcing Function

ωπ2

=T

ω

T

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

ωω

SDOF Dynamics 26

Undamped Harmonic Loading

β ωω

=LOADING FREQUENCY

Define Structure’s NATURAL FREQUENCY

( )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)Dynamic Magnifier

SDOF Dynamics 27

SDOF Dynamics 28

-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

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.

-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,

SDOF Dynamics 30

-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

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

-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,

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

-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,

SDOF Dynamics 33

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

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

ξω

=c

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

D pk

o=−

− +2

1 22 2 2

ξββ ξβ( ) ( )

C pk

o=−

− +1

1 2

2

2 2 2

ββ ξβ( ) ( )

SDOF Dynamics 36

Damped Harmonic Loading (5% Damping)

SDOF Dynamics 37

-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

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)

SDOF Dynamics 38

SDOF Dynamics 39

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

SDOF Dynamics 40

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 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

F(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

dt

dtdFF ττ =)(

dF

dt

τ

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 =

)(2

Obtain Exact Solution for next time step

τuu = )(2 τuu = )(2 τuu =

LOOP

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

∆t

LinearVelocity

QuadraticDisplacement

)(tu∆

)(tu∆

)(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

-0.40

-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

Earthquake Ground Motion - 1940 El Centro

Many ground motions now available via the Internet

SDOF Dynamics 55

-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

)

ut

Development of Effective Earthquake Force

ug ur

-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 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=

mk

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)

-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

The Elastic Displacement Response Spectrum

SDOF Dynamics 60

0

4

8

12

16

0 2 4 6 8 10

PERIOD, Seconds

DIS

PLA

CEM

ENT,

inch

es

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

-0.08

-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 SecondsUmax= 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 Displacement Response Spectrumfor El Centro Ground Motion

Elastic Response Spectrum

SDOF Dynamics 62


-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



SDOF Dynamics 63


-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



SDOF Dynamics 64


-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



SDOF Dynamics 65


-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



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

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



SDOF Dynamics 67

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

SDOF Dynamics 68

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

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

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

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

SDOF Dynamics 71

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

SDOF Dynamics 72

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

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

0.00

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, SecondsD

ispl

acem

ent,

Inch

es

@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

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

@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

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

SDOF Dynamics 77

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 per Second


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

SDOF Dynamics 78

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


SDOF Dynamics 79

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

SDOF Dynamics 80

Development of an ElasticResponse Spectrum

Problems With Current 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.

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

0.1

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

SDOF Dynamics 82

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

SDOF Dynamics 83

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

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

0.1

1

10

100

0.01 0.1 1 10 100Period (sec)

Disp

lace

men

t (in

)

0% Damping5% Damping10% Damping

Observations

gumax

gumax

gumax

gvv max→

gvv max→

0→v

0→v

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

0.1

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

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

1

3

2 4

65

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

Tf

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

3

2

1

“Short Period” Acceleration 1

3

2

S = 0.6 ST

T+0.4 SaDS

0DS

Spe

ctra

l Res

pons

e A

ccel

erat

ion,

Sa

SDS

S = Sa DS

S = ST

aD1

SD1

T = 0.2 SS

0D1

DS

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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