Earthquake Engineering 03-ResponseSpectra

8/11/2019 Earthquake Engineering 03-ResponseSpectra

1/48

January 28, 2003

1

Earthquake Engineering

GE / CE - 479/679

Topic 3. Response and Fourier Spectra

John G. Anderson

Professor of Geophysics


2/48

January 28, 2003

2

m

Earth

k

y0

F

yx = y-y0

(x is negative here)

Hookes Law

cFriction Law

xcF &=

kxF=

z(t)


3/48

January 28, 2003

3

In this case, the force acting on the mass dueto the spring and the dashpot is the same:

However, now the acceleration must bemeasured in an inertial reference frame,where the motion of the mass is (x(t)+z(t)).

In Newtons Second Law, this gives:

or:( )( ) xckxtztxm

&

&&

&&

=+)(

( )tzmkxxctxm &&&&& =++)(

xckxF &=


4/48

January 28, 2003

4

So, the differential equation for the forced

oscillator is:

After dividing by m, as previously, this equation becomes:

This is the differential equation that we use to characterize

both seismic instruments and as a simple approximation forsome structures, leading to the response spectrum.

( )tzxxhtx nn &&&&& =++2

2)(

( )tzmkxxctxm &&&&& =++)(


5/48

January 28, 2003

5

DuHammels Integral

( ) ( )

( )( ) ( ) ( )( )

dthth

h

tHatx nn

n

=

exp1sin

1

)(0

21

2

21

2

( ) ( )tzta &&=

This integral gives a general solution for the response of

the SDF oscillator. Let:

The response of the oscillator to a(t) is:


6/48

January 28, 2003

6

( )

( )( ) ( ) ( )( )thth

h

tHtx nn

n

= exp1sin

1

)( 21

2

21

2

Lets take the DuHammels integral apart to

understand it. First, consider the response of theoscillator to a(t) when a(t) is an impulse at time t=0.

Model this by:

( ) ( )tta =

The result is:


7/48

January 28, 2003

7

H(t) is the Heaviside step function. It is defined as:

H(t)=0, t=0

This removes any acausal part of the solution - the

oscillator starts only when the input arrives.

t=0

0

1


8/48

January 28, 2003

8

This is the result for an oscillator with fn= 1.0 Hz and h=0.05.

It is the same as the result for the free, damped oscillator with

initial conditions of zero displacement but positive velocity.


9/48

January 28, 2003

9

( ) ( )

( )( ) ( ) ( )( )

dthth

h

tHatx nn

n

=

exp1sin

1

)(0

21

2

21

2

The complete integral can be regarded as the result of summing

the contributions from many impulses.

The ground motion a() can be regarded as

an envelope of numerous impulses, each

with its own time delay and amplitude.The delay of each impulse is . The

argument (t- ) in the response gives

response to the impulse delayed to the

proper start time. The integral sums up all

the contributions.


10/48

January 28, 2003

10

Convolutions.

In general, an integral of the form

( ) ( ) dtbatx

=0

)(

is known as a convolution. The properties of convolutions

have been studied extensively by mathematicians.


11/48

January 28, 2003

11


12/48

January 28, 2003

12

Examples

How do oscillators with different dampingrespond to the same record?

Seismologists prefer high damping, i.e.

h~0.8-1.0.

Structures generally have low damping, i.e.

h~0.01-0.2.


13/48

January 28, 2003

13


14/48

January 28, 2003

14


15/48

January 28, 2003

15


16/48

January 28, 2003

16

Response Spectra

The response of an oscillator to an inputaccelerogram can be considered a simple

example of the response of a structure. It is

useful to be able to characterize anaccelerogram by the response of many

different structures with different naturalfrequencies. That is the purpose of the

response spectra.


17/48

January 28, 2003

17

What is a Spectrum?

A spectrum is, first of all, a function offrequency.

Second, for our purposes, it is determined

from a single time series, such as a recordof the ground motion.

The spectrum in general shows somefrequency-dependent characteristic of theground motion.


18/48

January 28, 2003

18

Displacement Response Spectrum

Consider a suite of several SDF oscillators.

They all have the same damping h (e.g. h=0.05)

They each have a different natural frequency fn.

They each respond somewhat differently to the

same earthquake record.

Generate the displacement response, x(t) for each.


19/48

January 28, 2003

19


20/48

January 28, 2003

20

Use these calculations to form thedisplacement response spectrum.

Measure the maximum excursion of each

oscillator from zero.

Plot that maximum excursion as a functionof the natural frequency of the oscillator, fn.

One may also plot that maximum excursionas a function of the natural period of the

oscillator, Tn

=1/fn

.


21/48

January 28, 2003

21


22/48

January 28, 2003

22

Definition

Displacement Response Spectrum. Designate by SD.

SD can be a function of either frequency orperiod.


23/48

January 28, 2003

23

Assymptotic properties

Follow from the equation of motion

Suppose n is very small --> 0. Thenapproximately,

( )tzxxhtx nn &&&&& =++22)(

( )tztx &&&& =)(

So at low frequencies, x(t)=z(t), so SD is

asymptotic to the peak displacement of the

ground.


24/48

January 28, 2003

24

Assymptotic properties

Follow from the equation of motion

Suppose n is very large. Then approximately,

( )tzxxhtx nn &&&&& =++22)(

So at high frequencies, SD is asymptotic to thepeak acceleration of the ground divided by the

angular frequency.

( )tzxn &&=2


25/48

January 28, 2003

25

Velocity Response Spectrum

Consider a suite of several SDF oscillators.

They all have the same damping h (e.g.h=0.05)

They each have a different naturalfrequency fn.

They each respond somewhat differently tothe same earthquake record.

Generate the velocity response, for each.( )tx&


26/48

January 28, 2003

26


27/48

January 28, 2003

27

Use these calculations to form thevelocity response spectrum.

Measure the maximum velocity of each

oscillator.

Plot that maximum velocity as a function ofthe natural frequency of the oscillator, fn.

One may also plot that maximum velocityas a function of the natural period of the

oscillator, Tn

=1/fn

.


28/48

January 28, 2003

28


29/48

January 28, 2003

29

Definition

Velocity Response Spectrum. Designate by SV.

SV can be a function of either frequency or

period.


30/48

January 28, 2003

30

How is SD related to SV?

Consider first a sinusoidal function: The velocity will be:

Seismograms and the response of structures

are not perfectly sinusiodal. Nevertheless, this

is a useful approximation.

We define:

And we recognize that:

( ) ( )ttx nn cos=&

SDPSV n=

( ) ( )ttx n

sin=

SVPSV


31/48

January 28, 2003

31

Definition

PSV is the Pseudo-relative velocityspectrum

The definition is: SDPSV n=


32/48

January 28, 2003

32


33/48

January 28, 2003

33

PSV plot discussion

This PSV spectrum is plotted on tripartiteaxes.

The axes that slope down to the right can be

used to read SD directly.

The axes that slope up to the right can be

used to read PSA directly. The definition of PSA is SDPSA n

2=


34/48

January 28, 2003

34

PSV plot discussion

This PSV spectrum shows results forseveral different dampings all at once.

In general, for a higher damping, the

spectral values decrease.


35/48

January 28, 2003

35

PSV plot discussion

Considering the asymptotic properties of

SD, you can read the peak displacement and

the peak acceleration of the record directly

from this plot.

Peak acceleration ~ 0.1g

Peak displacement ~ 0.03 cm


36/48

January 28, 2003

36

Absolute Acceleration Response, SA

One more kind of response spectrum.

This one is derived from the equations of motion:

( )tzxxhtx nn &&&&& =++22)(

SA is the maximum acceleration of the mass in aninertial frame of reference:

This can be rearranged as follows:

( ) xxhtztx nn22)( =+ &&&&&

( ){ }tztxSA &&&& += )(max


37/48

January 28, 2003

37

Summary: 5 types of response spectra

SD = Maximum relative displacement

response.

SV = Maximum relative velocity response.

SA = Maximum absolute acceleration

response

SDPSA n2

=

SDPSV n=


38/48

January 28, 2003

38

H l f


39/48

January 28, 2003

39

Here are some more examples of

response spectra

Magnitude dependence at fixed distance

from a ground motion prediction model, aka

regression.

Distance dependence at fixed magnitude

from a ground motion prediction model, aka

regression. Data from Guerrero, Mexico.

Data from Guerrero, Mexico, Anderson and Quaas (1988)


40/48

January 28, 2003

40


41/48

January 28, 2003

41


42/48

January 28, 2003

42


43/48

January 28, 2003

43


44/48

January 28, 2003

44

Main Point from these spectra

Magnitude dependence.

High frequencies increase slowly with

magnitude.

Low frequencies increase much faster withmagnitude.


45/48

January 28, 2003

45


46/48

January 28, 2003

46

Note about regressions

Smoother than any individual data.

Magnitude dependence may be

underestimated.


47/48

January 28, 2003

47


48/48

January 28, 2003

48

Note about regressions

Spectral amplitudes decrease with distance.

High frequencies decrease more rapidly

with distance.

Low frequencies decrease less rapidly.

This feature of the distance dependence

makes good physical sense.

Earthquake Engineering 03-ResponseSpectra

Documents

Transcript of Earthquake Engineering 03-ResponseSpectra