Response Spectra

Objectives

1. Define a response spectrum.

2. Give the uses of a response spectrum.

3. Define the types of response spectra.

4. Know how to calculate a velocity and acceleration time history from adisplacement response spectrum.

5. Know how to calculate a deterministic response spectrum for a given earthquakemagnitude and source distance.

6. Know how to calculate a IBC 2000 spectrum.

Figure 1. Displacement response spectrum for 2 percentdamping (after Chopra).

1. Definition

The peak or maximum response (acceleration, velocity, displacement) of all possible linear singledegree of freedom (SDF) system to a particular component of ground motion for a given level ofdamping.

2. Uses of the response spectrum

a. The response spectrum provided a convenient and practical way to summarize thefrequency content of a given acceleration, velocity or displacement time history.

b. It provides a practical way to apply the knowledge of structural dynamics to designof structures and development of lateral force requirements in building codes.

Response Spectra (5% Damping)

Spe

ctra

l Acc

eler

atio

n (g

)

Period (sec)

0.00

0.05

0.10

0.15

0.20

0.25

0 1 2 3 4 5

Acceleration Time History

Acc

eler

atio

n (g

)

Time (sec)

-0.02

-0.04

-0.06

-0.08

0.00

0.02

0.04

0.06

0 10 20 30 40

Figure 2. Acceleration Response Spectrum for Yerba Buena Record.

Figure 3. Acceleration Time History for Yerba Buena Island from the 1989 Loma PrietaEarthquake.

3. Types of Response Spectra

a. Response Spectra calculated from actual time histories

Response Spectra (5% Damping)

Spe

ctra

l Dis

plac

emen

t (ft)

Period (sec)

0.0

0.1

0.2

0.3

0.4

0 1 2 3 4 5

Response Spectra (5% Damping)

Spe

ctra

l Vel

ocity

(ft/

sec)

Period (sec)

0.0

0.2

0.4

0.6

0.8

0 1 2 3 4 5

Figure 4. Displacement Response Spectrum for the Yerba Buena Record.

Figure 5. Velocity Response Spectrum for the Yerba Buena Record.

Steps for calculating a response spectrum from a time history

The response spectrum for a given ground motion component (e.g., a(t)) is developedusing the following steps:

(1) Obtain the ground motion (a(t)) for an earthquake. Typically the accelerationvalues should be defined at time steps of 0.02 second, or less.

(2) Select the natural vibration period, T , and damping ratio, ξ, for SDOF system. n

(Usually 5 percent damping is selected.)

(3) Determine the maximum displacement response for a SDOF structure with theselected percent damping for a given period or frequency of vibration.

To do this, you must solve the following differential equation

mM u/dt + c Mu/dt + k u = F(t)2 2

This can be solved using (see Chopra, Ch. 1, pp. 28-32):

(a) Closed form solution (linear systems)

! Solution is only valid for initial conditions at rest

(b) Duhamel’s integral (linear systems)

! based on treating the periodic motion as a series of shortimpulses

(c) Frequency Domain method (linear systems)

! Fourier transform

! Inverse Fourier transform

(d) Numerical methods (linear and nonlinear systems)

! Numerical time stepping methods

(4) Repeat step 3 and vary the fundamental period of the structure by changing the mass(m), the stiffness (k), or both. Plot the new results.

Example of Using NONLIN to create a displacement response spectrum

First linear analysis

m = 100 kipsk = 100 kips / inc = 5 percent of critical dampingearthquake record is Imperial Valley El Centro Record (Impval1.acc)

period = 0.32 s, f = 3.13 Hz, ω = 19.65 rad/s

max. displacement = 0.700 inches

Second linear analysis

m = 100 kipsk = 50 kips / inc = 5 percent of critical dampingearthquake record is Imperial Valley El Centro Record (Impval1.acc)

period = 0.45 s, f = 2.21 Hz, ω = 13.89 rad/s

max. displacement = 1.677 inches

Third linear analysis

m = 100 kipsk = 25 kips / inc = 5 percent of critical dampingearthquake record is Imperial Valley El Centro Record (Impval1.acc)

period = 0.64 s, f = 1.56 Hz, ω = 9.82 rad/s

max. displacement = 3.051 inches

Example of Using NONLIN to create a displacement response spectrum

Fourth linear analysis

m = 100 kipsk = 200 kips / inc = 5 percent of critical dampingearthquake record is Imperial Valley El Centro Record (Impval1.acc)

period = 0.23 s, f = 4.42 Hz, ω = 27.79 rad/s

max. displacement = 0.359 inches

Fifth linear analysis

m = 100 kipsk = 400 kips / inc = 5 percent of critical dampingearthquake record is Imperial Valley El Centro Record (Impval1.acc)

period = 0.16 s, f = 6.25 Hz, ω = 39.30 rad/s

max. displacement = 0.135 inches

Sixth linear analysis

m = 100 kipsk = 1000 kips / inc = 5 percent of critical dampingearthquake record is Imperial Valley El Centro Record (Impval1.acc)

period = 0.10 s, f =0.259 Hz, ω = 62.14 rad/s

max. displacement = 0.056 inches

Displacement Response Spectrum

00.5

11.5

22.5

33.5

44.5

5

0 0.2 0.4 0.6 0.8

Period (s)

Max

. Dis

pla

cem

ent

(in

)

Figure 6. Displacement Response Spectrum for El CentroRecord.

Figure 7. Displacement Response Spectrum fromNONLIN.

Velocity Response Spectrum

0

5

10

15

20

25

30

35

0 0.5 1

Period (s)

Max

. Pse

ud

o V

elo

city

(in

/s)

Figure 8. Velocity Response Spectrum for example.

Figure 9. Velocity Response Spectrum from NONLIN

(5) Calculate the other pseudospectral velocity values using:

V = ωω D

Acceleration Response Spectrum

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8

Period (s)

Max

. Pse

ud

o A

ccel

erat

ion

(g

)

Figure 10. Acceleration Response Spectrum

Figure 11. Acceleration Response Spectrumfrom NONLIN

(6) Calculate the pseudoacceleration response values from:

A = ωω D2

It is common to express A in units of g, thus:

A(in g) = ωω D/g2

C o m p a r i s o n R e s p o n s e S p e c t r a

0.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.61.71.81.92.0

0 0.5 1 1.5 2 2.5 3

Period (s)

Sp

ectr

al a

ccel

erat

ion

(g

)

A b ra h a m s o n a n d S ilv a ( 1 9 9 7 ) R o c k o r S tiff S o il

B o o re , J o y n e r a n d F u m a l ( 1 9 9 7 ) V s = 7 6 2

S p u d ic h e t a l . ( 1 9 9 9 ) R o c k

Figure 12. Comparison of Deterministic Rock Spectra for M =7.0, R = 5 km earthquake from attenuation relations.

b. Deterministic Response Spectra from attenuation relations

! Deterministic spectrum are usually developed for the maximum credibleearthquake (MCE).

! The maximum credible earthquake is the largest earthquake possible from theactive faults in the region.

55 . 5

66 . 5

77 . 5 2 5

5 07 5

1 0 01 2 5

1 5 0

0

1 0

2 0

3 0

4 0

5 0

6 0

7 0

% Contr ibut io

n

M a g n i t u d e ( M ) D is tance (km)

D e a g g r e g a t e d S e i s m i c H a z a r d f o r P G A2 5 0 0 r e t u r n p e r i o d y e a r e v e n t

S a l t L a k e C i t y , U t a h ( S o u r c e : U S G S )pga = 0 .815g

Uniform Hazard Spectra (Rock) for Salt Lake City

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.5 1 1.5

Period (s)

Sp

ectr

al A

ccel

erat

ion

(g

)

10 % PE in 50 yr

5 % PE in 50 yr

2 % PE in 50 yr

Figure 13. Seismic Hazard and Percent Contribution ofM and R pairs for 2500 year return period event (2percent probability of exceedance in 50 years) for peakground acceleration.

Figure 14. Probabilistic Uniform Hazard Spectra for inputzipcode 84115 for USGS.

c. Probabilistic Spectra for Probabilistic Seismic Hazard Analysis (PSHA)

d. Design Spectra Developed from Building Codes

(See Lecture 6b).