Are We Ready for ELcentro

8/8/2019 Are We Ready for ELcentro

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Are We Ready for . . . "El Centro"?

by Bobby Motwani

03/03/10

(

http://bobbyness.net/NerdyStuff/structures/El%20Centro%20in%20Malaga/ElCentroIn

Malaga.html )

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right-click and save or open in separate window for better viewing.

y Objectivesy Softwarey Part I : El Centro

o Accelerogramo Time Response of SDOF systemso Response Spectrum

y Part II : Design Spectrao Response Spectra Characteristics

Time integration and Response Spectrum Calculation Revisitedo Design Spectra based on Response Spectrao Seismic Hazard Analysiso Uniform Hazard Spectrao Design Codes

EC8 NCSR-02

y Are we ready for El Centro?y References

Objectives:

The end goal of this study is fairly straightforward: I wish to determine how the buildings

designed to withstand seismic action in Malaga City would cope if the area were struck by

the historical earthquake that hit the city of El Centro, in the Imperial Valley in California, in

1940. More precisely, to compare the design-spectra specified by the Spanish Seismic

Building Design Code (currently the NCSE-02) with the response-spectra of the El Centro

Earthquake. I shall restrict this study to the effects of horizontal ground motion, and work

only with the N-S component of the El Centro ground motion by way of example (you could

carry out a similar study with the E-W component).T

hus, the study seems fairly simple: all Ineed to do is superimpose the N-S response-spectrum of the earthquake to the horizontal-

motion design-spectrum specified by the design code, and see whether the former lies

'within' the latter.

And that is precisely what I shall do. However, my ulterior motives for carrying out this

study are: a) to show how to compute response spectra of ground motion from data

collected by seismographs in the form of accelerograms, and b) to understand how design

spectra are constructed and to familiarize myself a bit with the Spanish Seismic Design

Code and the respective European Standards from which it is derived (Eurocode 8 ) and


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with the design criteria adopted in these codes. In the process we shall also learn about the

nature of response spectra, the historical design spectra by Newmark and Hall, and a

primer on Seismic Hazard Analysis from which the concept of Uniform Hazard Spectra

derives.

Software:

To carry out the required mathematical computations and plot data, I have used Matlab. Ihave deliberately used a general purpose numerical computing environment because:

a) part of the objective of this study is to learn to construct the required diagrams and not

rely on specialized software that does the job for you at the click of a button, which is not a

bad thing as long as you understand what is is the programme is doing to be able to

interpret the outputs correctly, but as I said, it's instructive to programme the process

yourself at least once! and,

b) for the flexibility this offers in handling data and carrying out operations to measure

rather than adhere yourself to the limited options a specialized programme provides.

I might also add that while the Matlab package is not available free of charge, you could just

as well do the same using other numerical computing environments such as Octave, Scilab,

or Rlab, which are available at no cost at all, or use a general purpose programming

language such as C/C++ or VB.NET (although with these I assume you'd have to develop

your own routines for solving differential equations and such, b ut those are topics

concerning numerical methods which are not the object of the present work!)

Part I : El Centro.

1. The accelerogram

The accelerogram for El Centro is provided by the NISEE (National Information Service for

Earthquake Engineering) supported by the University of California, Bekerley. The data is

provided as an ASCII-file (IMPVAL1.ACC) as a discrete series of readings. Ground

acceleration, velocity, and displacement are each provided in the same file. To simplify

reading from file into a MATLAB variable, I have separated each of the three sets of data

into separate text files (IMPVAL1-acc.txt, IMPVAL1-vel.txt, IMPVAL1-disp.txt), though I

shall primarily use the acceleration data.

To read the acceleration data from MATLAB, you can use the File I/O function "fscanf",

which reads and converts data from a text file into an array in column order. Theinstructions are in the m-file ElCentroAccelerogram.m :


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El

entroAccelerogram.m

Th earth uake durati is 53.76 seconds (2688 acceleration data pointsspaced evenly every 0.02

seconds - note thatthe first data point is fort=0.02s;the acceleration fort=0 is 0). Exectuting the

previous m-file givesus the plotofthe acceleration data:

horizontal ground acceleration of El

entro earthquake

2.

i

p

SDOF systems

The time response of a linear SDOF system to a ground acceleration is governed by thedifferential equation:

% El Centr Earthquake,

% plt acceler

gram

t=0:0.02:53.76;

fid = f pen('C:\Users\u\D

acc.txt');

a = fscanf(fid, '%g');

a = a./9800; % acceler

a = [0 ; a] % add data

pl

t(t,a,'c

l

r','b');gr

Title('IMPERIAL VALLEY E

ylabel('accelerati n (g)

xlabel('time (s)')


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wheregisthe ground acceleration, and uthe displacementrelative to the ground. We can

write this as:

orin terms ofthe natural period of vibration as:

To integrate this numerically in the time domain, Iused MATLAB's ode45 solver; the odesolversrequire the differential equation to be defined as a firstordersystem of ode'sin a

seperate m file as a function that returns the value of the derivative vector at a specifiedtime based on the value of the function calculated in the previous step of the integration.Other parameters can be passed on to this function, and since Ishall be obtaining the time

response for differentSDOFsystems, itmakessense to passthe damping ratio and naturalperiod of vibration to this function as parameters; it is also necessary to pass as aparamater the forcing function (ground acceleration), which I have passed as the pair of

vectors (acceleration vectorand corresponding time vector), so thatthe acceleration attherequired time instant can be computed as necessary (via a linear interpolation, forexample). The m-file containing the equationsis available asf.m:

f.m

sing this function, we can, for example, simulate the free response of a SDOF system- eg, for a

SDOFsystem with a 5% damping ratioand natural period of vibration of 10s:

functinyd

t = f(t,y,p1,p2,

xi = p1;

Tn= p2;

t1 = p3; % time vectr

a2 = p4; % gr

und accelerat

a = interp1(t1,a2,t); % Int

ydt = [y(2); -(4*pi*xi/Tn)

% Time Respnse

f

%eg free respnse

xi=0.05; % relativTn=10; % natural

a0 = zer

s(length(

y0 = [1;0]; % ini

[T,Y] = de45(@f,t

Y(:,2) in m/s

plt(T,Y(:,1));gri

Title('SDOF (5% da

displacement')

AXIS tight

ylabel('displaceme

xlabel('time (s)')


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TimeRespo seSDOFSystems.m

free respo

se ofSDOFsystem with 5%dampingandnat

ralperiodof10s

To simulate the time-response to the ground acceleration, I have first augmented the simulation

time by 50% of the earthquake duration, just in case the peak value of the response (which will be

the value of interest to us) occurs during the natural vibration of the system that will follow the

extinction of the ground motion. I have then used ode45 to obtain the steady-state response of the

system to the ground acceleration, and plotted the system deformation (displacement of the mass

relative to the ground) as a function of time:


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TimeResponse

O

ystems.m

time response of

O

system with 5% damping and natural period of10s to El entro ground motion

% incre

earthqu

t1 = 0:

a2 = a*

diff =

a2 = [a

% stead

y0 = [0

[T,Y] =

%pl ! t !

subpl!t

pl ! t(t1

grid

Title('

axis ti

ylabel(

xlabel(

%pl ! t !

subpl ! t

pl ! t(T,

Title('

ylabel(

xlabel(

axis ti

%maximu

max(abs


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The maximum displacement of the mass relative to the ground is seen to be, in this case,

38.41 cm.

3. Response Spectrum

If we repeat the previous analysis for different SDOF systems, maintaining the dampingratio and varying the natural period of vibration, and represent the maximum value of each

response against the period, we obtain the Displacement (or Deformation) Response

Spectrum of the El Centro Earthquake for a particular damping ratio (it's customary to use

a damping ratio of 5%, which is the normal damping ratio of most buildings):

ResponseSpectra.m


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ElCentroDeformation Response Spectr" mfor5% dampingratio

In logarithmic axes:


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ElCentroDeformation Response Spectr# mfor5%dampingratio

It is also of interest to plot the pseudo-velocity and pseudo-acceleration response spectra.

The pseudo-velocity and pseudo-acceleration are given by:

that is,

The respective plots are: (the MATLAB code for this section can be found in the m-file

ResponseSpectra.m)


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Pseudo-velocityresponse spectrumforElCentroEarthquake, 5%dampingratio

Pseudo-$

cceleration response spectrumforElCentroEarthquake, 5%dampingratio

It is also of interest to plot normalised versions of these spectra, that is, normalised w.r.t to

maximum ground displacement, velocity, and acceleration. I shall represent the normalised

displacement spectrum and normalised acceleration spectrum, as these are of particular interest (I

shall why explain in a moment) ie: D/ug0 andA/g0 :


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Relative Deformation response spectrumforElCentroEarthquake, 5%dampingratio


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Relative Acceleration response spectrumforElCentroEarthquake, 5%dampingratio

Although the deformation, pseudo-velocity, and pseudo-acceleration response spectra

contain the same information, it is useful to plot all three spectra because each one directly

provides a meaningful physical quantity: the deformation spectrum provides the peak

deformation of a system, the pseudo-velocity is directly related to the peak strain energy

stored in the system during the earthquake, and the pseudo-acceleration times the mass is

equal to the peak equivalent static force and base shear (see Chopra).

Finally, it is also of interest to plot all three spectra (deformation, pseudo-velocity, and

pseudo-acceleration) on the same graph. This is called a tripartite plot, and is possible

because all three separate quantities contain the same information, no more and no less.

Basically, we start off by plotting V against the period (or frequency) on logarithmically

scaled axes, as we did earlier. Now, since:

we see that lines of constant D have a slope of -1 on logarithmically spaced axes, hence, the

axis of deformation has a slope of 1. Similarly, noting that:

that is, lines of constantA have a slope of +1 on logarithmically spaced axes, hence, the axis

of pseudo-acceleration has a slope of -1. Thus, we have the two mutually perpendicular

vertical and horizontal axes for V and T, and another two mutually perpendicular axes for D

and A which for angles of 45 w.r.t. the vertical and horizontal axes. That is why this type of

plot is known as a four way log plot.


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Note that for T 2 , A V D, as can be readily seen from the relations

To represent the three response spectra on a single 4-way log plot, all we need to do is

represent V againstT and then plot lines of constant D and lines of constant A. The file

ResponseSpectra.m contains the code I used to generate these lines and their labelling.


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Tripartite plotofElCentro Response Spectrum.(Green lines: pseudo-acceleration lines;maroon lines: constantdisplacementlines).

Part II: Design Spectra

1. Response Spectra Characteristics


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The main reason for plotting all three response spectra on the same plot (tripartite plot)

rather than on seperate plots is because the shape of the spectrum can be approximated

more readily for design purposes with the aid of all three spectral quantities than with one

alone. Let's have another look at the response spectrum for El Centro, for a damping ratio of

5%. This time I plot a larger range of natural vibration periods (Tn 0.01:0.1:50.1) :

Tripartite plotofElCentro Response Spectrum(blue), maximumgroundmotions(red)

I have also plotted in red the peak ground motion values: the peak ground displacement as

constant displacement line, the peak ground velocity as a constant pseudo -velocity line, and

the peak ground acceleration as a constant pseudo-acceleration line.This is to compare the

three spectral response quantities (displacement, pseudo-velocity, and pseudo-


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acceleration) with the respective peak ground motion maxima (note though that peak

pseudo-velocity and pseudo-acceleration are notthe peak relative velocity and acceleration

of the system, so strictly speaking it doesn't make sense to plot peak ground velocity and

peak ground acceleration on the same scales as system response peak pseudo-velocity and

peak pseudo-acceleration respectively, at least not on their own, - however, I can only

assume that this is done to normalize the pseudoquantities with respect to ground maxima

(which is okay as we are really only plotting normalised pseudoquantities), which gives us

a means to construct smooth design spectra from peak ground motion parameters based onthe trends in the response spectra which are observed, as we shall see in what follows).

When we observe response spectra such as that plotted for El Centro in the above plot, for

different site readings of earthquake motion or different damping ratios, we notice the

following trends:

y for systems with very short natural periods of vibration, the peak pseudo-acceleration (A) approaches g0 and the relative displacement (D) is very small.

Why? Systems with very short periods are very rigid with relatively very little

inertia, thus the mass moves rigidly with the ground. As a consequence,

o D is very small, since D is deformation or relative displacement w.r.t. grounddisplacement, and

o relative acceleration ((t)) will be nearly zero, which implies thatg(t) t(t).For lightly damped systems, it can be shown thatt(t) -A(t) (see Chopra),

thereforeA g0.


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Response spectraofElCentroforvariousdampingratios,figure borrowedfromChopra.

for systems with very long natural periods of vibration (usually above 15s or so), D

approaches ug0, and A is very small (since the forces in the structura are directly related to

mA, these are small too). The reasoning is as follows: systems with very long natural

periods are very flexible with relatively large inertia (inertia >> rigidity), therefore, the

mass remains almost stationary as the ground below it moves. Hence:

y t(t) 0 A(t) 0, andy u(t) - ug(t) D ug0.


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Response spectraofElCentroforvariousdampingratios,figure borrowedfrom Chopra.

To be honest, my plot of the response spectrum for a damping ratio of 5% doesn't

converge to the peak ground displacement, at least not for natural periods of 50s or

less. Initially I thought that constructing the response spectrum for even longer

periods (of little practical interest as far as civil structural design goes) would

eventually lead to this convergence, but the results for the same response spectrum

presented by other authors (such as the one I borrowed from Chopra to illustrate

the trends, as shown above), show that the maximum relative displacement D, (for

different damping ratios too) tends to the maximum ground displacement long

before 50s (clearly not a fault of the different units being used). My results seem to

be OK though for smaller periods, as I have contrasted them with those provided by

other sources - I can only assume that the ode solver I have used of MATLAB is notproducing accurate results for large periods, maybe because the equations for such

periods require other specialized solvers to avoid growth of numerical errors and

such, but that's a topic I shall not go into. I shall just pretend, for this study, that my

plot follows, approximately, the trends I have described above, a summary of which I

show in the following figure:


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Trends in Response Spectrarelative togroundmotion parameters, borrowedfrom Chopra.

From the previous plot it can be observed that response spectra tend to be characterized by

three regions:

y a region of approximately constant acceleration, in the high-frequency (shortperiod) portion of the spectra: this region is called the acceleration sensitive region

because structural response seems to be most directly related to ground

acceleration

y constant displacement, at low frequencies (long-period region): this region is calledthe displacementsensitive region as structural response seems to be related most

directly to ground displacement

y constant velocity, at intermediate frequencies: this region is called the velocity-sensitive region, because structural response seems to be better related to ground

velocity than to other ground motion parameters.

In each region, the corresponding ground motion is amplified the most (although for very

low and very high periods, the amplification tends to unity, as we have reasoned). (NB,

although I used the term 'amplified', you should bear in mind that it is the pseudo -spectral

quantities that we are viewing as amplifications of ground maxima. Remember that pseduo-

acceleration and pseudo-velocity are not the peak acceleration and peak velocity of the

system response respectively, thus, to be strict, the amplification factors are not showing


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how peak ground velocity and ground acceleration are amplified. However, it seems that

the psedo-velocity and pseudo-acceleration are good approximations of relative velocity

and absolute acceleration in certain cases).

Regions ofthe response spectra,graphborrowedfrom Chopra.

1b. Time integration and Response Spectrum Calculation Revisited

I've decided to try and improve on two aspects of my procedure so far that I'm not too

happy with. The first, to use another method of integrating the system response in the timedomain, in view of my uncertainty of MATLAB's ode45's performance for high periods. This

will give me a means to contrast the results obtained up till now.

The proposed method of integration of the dynamic equation is the "Piecewise exact

method" (see FEMA notes). Basically, the loading function is appromiated by a series of

straight-line segments, and then, within each of these intervals, the dynamic equation is

solved (rather, the analytical expression of the solution for a ramp input, which is easily

obtainable, is evaluated) as a function of the values of the response at the start of the

interval (initial conditions). This gives us the initial conditions for the next segment. The


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deduction of the analytical equations of both the deformation and velocity to a ramp

loading function are shown in the document piecewiseMethod.jpg. The MATLAB file in

which I've implemented the method is piecewiseIntegration.m.

Analyticalsolution ofdynamic equation toramp loading.


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piecewiseIntegration.m.

It's advantageous to use this method because even though it is an approximate method due

to the fact that the loading function is approximated by a series of straight-line segments, in

our case, the loading function is given by a series of discrete points, which essentially is a

series of straight-line segments (it is the most precise data we have of the load anyway). So

barring the precision of the accelerogram data, the method is exact, because we are directlyevaluating the analytical expressions solution to the differential eqution. And it's far less

costly from a compatitonal point of view than using ode45, which is based on a four and five

stage Runge-Kutta method for numerical integration of odes.

The response spectrum, constructed using piecewise integration to solve the dynamic

equation for different natural periods, is shown in the next figure (the code details can be

seen in the m-file ResponseSpectrumUsingPiecewiseIntegration.m


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Deformation Response SpectrumforElCentro, 5% damping, obtainedusing piecewise integration.

A scaled version, w.r.t. maximum ground motion, is shown next:


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ScaledDeformation Response SpectrumforElCentro, 5%damping, obtainedusing piecewise

integration.

The result is interesting, because even for a natural period of up to 50s, the deformation ofthe system doesn't taper down to that of the ground's, as I was half-expecting it to. The

amplification of the displacement is roughly the same as that obtained using the ode solver.

I shall take this behaviour to be correct then, despite my earlier scepticism, though I don't

know why it is so. Perhaps one needs to simulate extremely long periods for that trend to

be observed, in the particular case of El Centro.

Even though I have accepted the trend for long periods computed using ode45, as the

piecewise method has predicted the same trend, it is still interesting to compare the results

obtained using both methods. The following figure shows the response spectrum computed

using ode45 in red, and the response spectrum computed using the piecewise method, in

blue. For both, computations of maximum relative displacement were carried out every

0.02 seconds from T=0.01 to 50s.

Despite the piecewise method of integration not having thrown any light as to the trend

issue for long periods, I'm glad I decided to implement it, as the previous graph shows that

the results obtained by the ode solver seem to become somewhat haphazard especially for

longer periods. The piecewise method, which evaluates the system response from the

analytical expression of the solution to the dynamic equation, is 'exact' in this respect (as

far as the loading can be approximated by a series of straight-line segments, which is

actually all the data we have considering the accelerogram is given as a series of discrete


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points), and it does not show this haphazard behaviour. I shall henceforth use the piecewise

method, which by my previous arguments, I'd reason provides the more precise results.

Despite the piecewise method of integration not having thrown any light as to the trend

issue for long periods, I'm glad I decided to implement it, as the previous graph shows that

the results obtained by the ode solver seem to become somewhat haphazard especially for

longer periods. The piecewise method, which evaluates the system response from the

analytical expression of the solution to the dynamic equation, is 'exact' in this respect (asfar as the loading can be approximated by a series of straight-line segments, which is

actually all the data we have considering the accelerogram is given as a series of discrete

points), and it does not show this haphazard behaviour. I shall henceforth use the piecewise

method, which by my previous arguments, I'd reason provides the more precise results.

The second aspect I wanted to improve was on the selection of natural period times for the

construction of log-plots of the response spectrum. Notice that if the periods are evenly

spaced, the corresponding data points are not evenly spaced on a log plot, but 'bunched'

towards the right of the graph. Eg, if the previous plot of the El Centro Response Spectrum,

with data points plotted every 0.02 seconds, is plotted on logarithmic axes, we see that the

actual data points (in red) are not evenly spaced, and the left side of the graph is made up

mostly of few straight lines interpolating these data points which are far apart.

For plots of spectra on logarithmic axes, whose object is to magnify the information relative to

shorter periods as against the very large periods (note that for civil engineering purposes, the

natural periods of interest usually are shorter than a couple of seconds at the very most), it makes

much more sense to carry out the calculations at logarithmically spaced points as well. Thus, in

MATLAB if we define Tn as Tn = logspace(-2,1.7,500), we obtain 500 points logarithmically spaced

between 10-2=0.01 and 10 1.7 50s, which will appear linearly spaced on a logarithmic axis:


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and the pseudo-velocity response spectrum:


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where the red dots are actual data points.

And the tripartite:


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where the red dashed-lines indicate peak ground motion values.

2. Design Spectra based on Response Spectra

The jaggedness of a response spectrum, such as the one of El Centro, is a typical feature of

response spectra because they are for individual earthquakes recorded at particular sites

with particular soil characteristics and what not. We cannot use such a spectrum for design

purposes, because small variations in system frequency would produce very different

design demands. And the design would only be valid (optimised) for that particular

earthquake at that particular location.


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Jaggedness ofResponse Spectra,figure borrowedfrom FEMA.

In contrast to response spectra, design spectra need to be smooth, since the natural period of a civilengineering structure cannot be determined with that much accuracy, and design specification

should not be very sensitive to a small change in natural period. Based on our previous ob servations

on the characteristic shapes of response spectra when represented on four-way log plots (well not

ourobservations, theirs, but you know what I mean), Newmark and Hall decided that the design

spectrum should consist of a series of straight lines in each of the three 'amplification' zones, and

then two connecting lines to the A= g0 and D =ug0 regions that we saw are characteristic for very

short and very long periods respectively:


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Creatinga smooth spectrumfordesign.

Newmark and Hall plotted the response spectra of dozens of earthquakes that were recorded on

firm soil sites for the western United States. Each ground motion was normalised so that all had the

same peak acceleration, eg. g0= 1g.


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Response Spectraofdifferentearthquakes (forthe samedampingratio).

At each period, the mean value of spectral ordinates and the standard deviation were calculated,

and the corresponding spectra were obtained:

Mean andmean + 1 spectrafordampingratio = 5%.

he peak ground velocity and displacement are average values over the set of ground

motions (remember that each individual spectrum was normalised w.r.t. the same peak

ground acceleration). The values are (for peak ground acceleration of 1g):

Using these results, and based upong the shape-trends discussed earlier, Newmark and Hallproposed a simplified spectrum made up of straight lines, based on the previous ground

motion parameters:


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IdealisedNewmarkspectrum (shownforthe mean case),forsystems with 5%dampingratio.

The design spectra shown in red on the previous plot is for systems with a 5% damping ratio, andfor a 50% probability of exceedance. For different damping ratios, and different probabilities of

exceedance, different idealised spectra are obtained. However, the recommended period values are

Ta= 1/33s, Tb=1/8s, Te=10s, and Tf=33s for all design spectra, whereas the periods Tc and Td are

obtained by intersection of the lines bc, cd and de (which are the lines of constant acceleration Ag0,

Bu'g0, and Dug0, although plotted as scaled values on the previous graph) for each particular

idealised spectrum. The amplification factors, obtained carrying out the previous construction for

different damping ratios and for both the mean and mean-plus-one-standard-deviation are found to

be:


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AmplificationfactorsforNewmarkandHall's design response spectra(from NewmarkandHall, 1982).

The Newmark-Hall idealised response spectrum is an empirically derived elastic design

spectrum (as it's derived from elastic response spectra). To construct the Newmark-Hall

design spectrum for a given ground motion, we start off by plotting the peak ground motionvalues (if only the peak ground acceleration is available, the peak ground velocity and

displacement obtained from the average values of the records analysed by Newmark and

Hall, given by

can be used. Then all you have to do is look up the amplification factors from the table for

the corresponding damping ratio and level of exceedance probability desired, and plot the

constant pseudo-acceleration, pseudo-velocity and relative displacement lines between

Tb=1/8s and Te=10s (the intersections of these lines give you Tc and Td). For T 33s, weplot the lines A/g0= 1 and D/ug0= 1, respectively, and finally connect points a and b, and e

and f with transition lines.

Carrying out this process in MATLAB, using the El Centro Earthquake peak ground motion

values, for a 50% probability of exceedance, and systems with a 5% damping ratio, the

result is: (the code is avaliable in the m-file: NewmarkDesignSpectrum.m).


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NewmarkDesign SpectrumforElCentroGroundMotion, 50% exceedance probability, 5%damping

systems.

The values ofTc and Td (although obtained graphically in the previous plot), can easily be

determined analytically:


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The elastic design spectrum by Newmark and Hall provides a basis for calculating the

design force and deformation for SDOF systems to be designed to remain elastic. This basic

form (with some modifications) is now the basis for structural design in seismic regions

throughout the world (typically plotted against structural "period", the inverse of

frequency). A nominal level of damping is assumed (5% of critical damping).

Let's plot the Newmark Design Spectrum for El Centro ground motion parameters, for 5%

damping ratio, that we obtained in the last section, but instead as pseudo-accelertaion

against system period, on arithmetic scales. For each straight-line segment of the graph, the

pseudoacceleration is given by:

y for


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y for

y


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y for

y for

y fory

for


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y y for

y for

y for


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y for

y For


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If we plot this in MATLAB: (see file NewmarkDesignSpectrum.m)

Newmark's Design SpectrumforElCentroEarthquake, 50% exceedance probability, 5% dampingratio.

where I have only plotted to 2.5 seconds, so not all curves are shown. We can compare the El Centroresponse spectrum with the design spectrum for El Centro - we did this earlier on the triparterepresentation, but let's see how it looks on the pseudoacceleration-T arithmetic axes:


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ElCentro Response Spectrum (blue), NewmarkDesign SpectrumforElCentrofor50% exceedanceprobability.Dampingratio 5%.

3. Seismic Hazard Analysis

The Newmark design spectrum was developed from records on rock site. However, one

would expect that the site conditions influence the frequency content in the ground motion,

and therefore the spectral amplification factors would depend on them. Although studies

have been carried out and site dependent spectra have been obtained (amplification factorsfor different soil types), the Newmark design spectrum was developed for a specific

collection of ground motion records. Where no ground motion records are available, the

procedure developed by Newmark for constructing design spectra would be extremely

limited.

What has been done is that Seismic hazard models of sites are used to develop another kind

of design spectrum known as a Uniform Hazard Spectrum that is used today for design

purposes. I will briefly introduce the concepts of Seismic HazardAnalysis.

Seismic Hazard Analysis describes the potential for dangerous, earthquake-related natural

phenomena such as ground shaking, fault rupture, or soil liquefaction. It should not be

confused with Seismic riskanalysis, which assesses the probability of occurrence of losses(human, social, economic) associated with the seismic hazards. For example, a hazard

associated with earthquakes is ground shaking. The risk is structural collapse and, possibly,

loss of life.

Seismic Hazard Analysis includes the quantitative assesment of expected levels of ground

shaking at a given location due to future earthquakes in the surrounding region. It is based

on the development of a seismicitymodel for the location and size of future earthquakes in

the region, and a groundmotion model by which the level of shaking can be predicted at a

given site as a result of the earthquake scenarios predicted by the seismicity model. The


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integration of the two models provide us with a model for the expected levels of shaking at

the site of interest.

A seismicity model, or more precisely, a seismic-hazard source model, is a description of

the magnitude, expected locations, and frequency of all earthquakes that affect a given site.

It is based on geological studies and records of seismic sources (active fault lines of the

region), regional earthquake catalogues, instrumental recordings, and such. I will not go

into details, as it is an extremely complex area of study, but basically the model mustspecify a list of scenarios (seismic sources that would cause seismic hazard at the site of

study), quantifying the magnitude, location, and rate of each one of them.

Consideration ofthe differentsource types thatcouldaffectagiven site

in the developmentofa seismicitymodelforthe site.

The ground motion model must give estimations of the ground motions caused by

earthquakes. The most basic ground motion models give the ground motion level (which

may be any parameter that characterises the shaking, such as peak ground acceleration

(PGA), spectral acceleration (for a certain period), etc) as a function of magnitude and

distance of the earthquake, but more complex models include other parameters to allow for

different site types (eg, rock vs soil). Equations are developed by fitting analytical

expression to observations (or to synthetic data where observations are lacking).

Example ofan empiricalattenuation relationship.


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Illustration ofattenuation withdistance.

The attenuation relationships, which form part of the ground motion model, are derived for

a particular region of interest. It depends a lot on the geological features of the region.

Probabilistic Seismic Hazard Alalysis (PSHA).

A deterministic seismic hazard analysis (DSHA) would simply consider the main sources

that affect a site, determine their magnitudes, and determine the ground motion that each

one would provoke at the site (usi ng the attenuation relationships):


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Deterministic seismic hazardanalysis.

In a probabilistic analysis, a recurrence relationship (eg, frequency of earthquakes above a

certain magnitude) is introduced as is the uncertainty in each step of the process. Each of

the uncertainties are included in a probabilistic analysis, and the result is a seismic hazard

curve that relates the design motion parameter (eg PGA) to the probability of exceedance:


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Example ofHazardCurve: annualfrequencyofexceedencevs PeakG roundAcceleration.

The probabilistic seismic hazardprovides the annualexceedence probabilityforgivenground

motions in hazardcurves, andthis methodologyhas todaypracticallysubstitutedthe classical

deterministic methods thatdeterminegroundshakingfrom mappedfaults assuminga certain

rupture

Once a hazard curve has been developed, we can obtain a design ground motion. The

hazard curve represents values of the average annual rate of exceedance for any given

ground-motion value. Assuming that ground motions may be described by a Poisson

distribution over time, then, if we define N(t) as the number of events that take place in

time interval [0,t] (random variable), the probability of there being k number of ground

motion events in a time tis given by:

where is the average number of events that occur in one time unit ( tis the average

number of events that occur in a time t). Note that (which defines the probability density

function), is provided by the hazard curves as a function of the ground motion parameter.

The probability of no events in a time period 0 to t is thus:


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and the probability of there being at least one event in the time 0 to t is thus:

It follows that the average annual rate of exceedance corresponding to a probability P of at

least once exceedance within a time period t is given by:

For example, if a designer wished to design a dam for the ground motion that has only a

10% probability of being exceeded in a 50 year period, first they would compute the

corresponding average rate of exceedance that defines the poisson distribution which

would give this probability for at least one event to occur in t=50yrs:

(this corresponds to a return-period of 1/ 474.6 yrs):

T

he ground motion parameter (egP

GA

) would be taken from the seismic hazard curve forthe site of the dam:


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In other words, there is a 10% percent chance that an earthquake that would provoke a

PGA of more than 0.33g at the site of the dam will occur in 50 years. (The hazard curve tells

us that a PGA that exceeds 0.33g occurs on average 0.021 times per year, or once every 475

yrs (return period)). (Likewise, for example, there would be a 63.21% probability for at

least one ground motion ofPGA above 0.33g to occur at the site in a period of time of 475

years:

(note that a 10% probability of exceedance in 50yrs has the same return period (475 years)

(or equivalently, a frequency of exceedance of 1/475) as a 63.21% probability of

exceedance in 475 years. There are infinte combinations ofP and t for a given . The hazardcurve tells us that at the site in question, a frequency of exceedance of 1/475 events/year

corresponds to the frequency of there being a ground motion with PGA above 0.33g at that

site). For building design, a 50-year base line is used as this is estimated as the service life

of a typical building. For structures such as dams, a longer return period may be more

appropriate as the required service life might be much longer.

Once a ground motion level has been defined, it is a question of engineering techniques to

design a structure that can withstand the shaking.

Seismic Hazard Maps.

Seismic maps are a representation of seismic hazard levels on territorial maps. The maps

are typically expressed in terms of probability of exceeding a certain ground motion.

Hazard maps commonly specify a 10% chance of exceedance (90% chance of non-

exceedance) of some ground motion parameter for an exposure time of 50 years,

corresponding to a return period of 475 years (this return period is selected rather

arbitrarily). The SESAME project has developed a continental-scale hazard map for Europe:


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The Seismic HazardMap ofthe European -Mediterranean region, in terms ofpeakground

acceleration ata 10% probabilityofexceedance in 50yearsforstiffsoilconditions, published

in2003.

This map depicts Peak Ground Acceleration (PGA) with a 10% chance of exceedance in 50

years for a firm soil condition. PGA, a short -period ground motion parameter that is

proportional to force, is the most commonly mapped ground motion parameter becausecurrent building codes that include seismic provisions specify the horizontal force a

building should be able to withstand during an earthquake.

The 10% probability of exceedance in 50 years maps depict an annual probability of 1 in

475 of being exceeded each year. This level of ground shaking has been used for designing

buildings in high seismic areas. The maps for 10% probability of exceedance in 50 years

show ground motions that we do not think will be exceeded in the next 50 years. In fact,

there is a 90% chance that these ground motions will NOT be exceeded. This probability

level allows engineers to design buildings for larger ground motions than what we think

will occur during a 50-year interval, which will make buildings safer than if they were only

designed for the ground motions that we expect to occur in the next 50 years.

The purpose of representing earthquake actions in a seismic design code such as EC8 is to

circumvent the necessity of carrying out a site-specific seismic hazard analysis for every

engineering project in seismically active regions. For non-critical structures it is generally

considered sufficient to provide a zonation map indicating the levels of expected ground

motions throughout the region of applicability of the code and then to use the parameters

represented in these zonations, together with a classification of the near-surface geology, in

order to construct the elastic design response spectrum at any given site.

4. Uniform Hazard Spectra

If a series of seismic hazard curves is developed for a range of different parameters (e.g.,

PGA, 0.1 sec spectral acceleration (fo a given damping ratio), 0.2 sec spectral acceleration,

etc.) and values are extracted for a certain constant probability (say a 10% in 50 year

probability of exceedance, which determines a return period, in this case of 475 years),

then a design response spectrum may be created by plotting the parameter magnitudes vs

period. The following plot shows the first point on such a spectrum -- in this case, the PGA

with a 10% probability of being exceeded in 50 years:


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PGA (= g0) = 0.33g is the peakgroundacceleration thathas a probabilityof10% ofbeing

surpassed in 50years atthe site.

If we now repeat the process with the hazard curve that gives us, for different values of

spectral acceleration of SDOF systems with a natural period of 0.2s, the average frequency

of exceedance, at the site, of each of these spectral accelerations, :

PSA = 0.55g is the peakspectralacceleration ofa T=0. 2s (5% damped) system, thathas a

probabilityof10% ofbeing surpassed in 50years atthe site.

If the process is continued, the result is a complete elastic response spectrum. It is called a

uniform hazard spectrum (UHS) because each point on the spectrum has the same


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probability of being exceeded in the given period. These spectra could then be used in a

response spectrum analysis of the structure.

Uniform HazardSpectrum, 5% dampedsystems, 10% probofexceedance in 50years.

The UHS is considered an appropriate probabilistic representation of the basic earthquake

actions at a particular location. The UHS will often be an envelope of the spectra associated

with different sources of seismicity, with short-period ordinates controlled by nearby

moderate-magnitude earthquakes and the longer-period part of the spectrum dominated

by larger and more distant events. As a consequence, the motion represented by the UHS

may not be appropriate for artificial ground motion generation:

Itshouldbe possible togenerate arealistic earthquakegroundmotion thatmatches the small

orlarge earthquake. However, ageneratedearthquake thatmatches the UHSwouldbe

unrealistic (andpossiblytoodemanding).


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However, if the only parameter of interest to the engineer is the maximum acceleration that

the structure will experience in its fundamental mode of vibration, regardless of the origin

of this motion or any other of its features (such as duration), then the UHS is a perfectly

acceptable format for the representation of the earthquake actions.

5. Design Codes

Until the late 1980s, seismic design codes invariably presented a single zonation map,

usually for a return period of 475 years, showing values of a parameter that in essence was

the PGA. This value was used to anchor a spectral shape specified for the type of site,

usually defined by the nature of the surface geology, and thus obtain the elastic design

spectrum. In many codes, the ordinates could also be multiplied by an importance factor,

which would increase the spectral ordinates (and thereby the effective return period) for

the design of structures required to perform to a higher level under the expected

earthquake actions, either because of the consequences of damage (e.g. large occupancy or

toxic materials) or because the facility would need to remain operational in a post-

earthquake situation (e.g. fire station or hospital). A code spectrum constructed in this way

would almost never be a UHS. Even at zero period, where the spectral acceleration is equal

to PGA, the associated return period would often not be the target value of 475 years since

the hazard contours were simplified into zones with a single representativePGA value over

the entire area. More importantly, this spectral construction technique did not allow the

specification of seismic loads to account for the fact that the shape of response spectrum

varies with earthquake magnitude as well as with site classification, with the result that

even if the PGA anchor value was associated with the exact design return period, it is very

unlikely indeed that the spectral ordinates at different periods would have the same return

period (McGuire, 1977). Consequently, the objective of a UHS is not met by anchoring

spectral shapes to the zero-period acceleration.

Within the drafting committee for EC8 there were extensive discussions about how the

elastic design spectra should be constructed, with the final decision being an inelegant and

almost anachronistic compromise to remain with spectral shapes anchored only to PGA. In

order to reduce the divergence from the target UHS, however, the code introduced two

different sets of spectral shapes (for different site classes), one for the higher seismicity

areas of southern Europe (Type 1) and the other for adoption in the less active areas of

northern Europe (Type 2). The Type 1 spectrum is in effect anchored to earthquakes of

magnitude close to Ms ~7 whereas the Type 2 spectrum is appropriate to events of Ms 5.5.

At any location where the dominant earthquake event underlying the hazard is different

from one or other of these magnitudes, the spectrum will tend to diverge from the target

475-year UHS, especially at longer periods.

5.1 Eurocode 8 Elastic Design Spectra.

"Forthe purpose ofthis Eurocode, nationalterritories shallbe subdividedbythe

NationalAuthorities into seismic zones, dependingon the localhazard. Bydefinition,

the hazardwithin eachzone can be assumedtobe constant.

"Formostofthe applications ofthis Eurocode, the hazard is described in terms ofa

single parameter, i.e.the value ofth e reference peakgroundacceleration on rockor

firm soilagR. Additionalparameters requiredforspecific types ofstructures aregiven


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in the relevantParts ofEurocode 8.

"The reference peakgroundacceleration, chosen bythe NationalAuthoritiesforeach

seismic zone, corresponds tothe reference return periodchosen byNational

Authorities. Tothis reference return periodan importancefactor1 equalto 1.0 is

assigned.Forreturn periods otherthan the reference the designgroundacceleration

ag is equaltoagR times the importancefactor1.

"Within the scope ofEN1998 the earthquake motion atagiven pointofthe surface is

generallyrepresentedbyan elasticgroundacceleration response spectrum, henceforth

calledelastic response spectrum

"Twodifferentshapes ofresponse spectra, Type 1 andType2 maybe adopted. The

NationalAuthoritymustdecide which elastic response spectrum, Type 1 or/andType

2, toadoptfortheirnationalterritoryorpartthereof. In selectingthe appropriate

spectrum, consideration shouldbegiven tothe magnitude ofearthquakes thataffect

the nationalterritoryorpartthereof. Ifthe largestearthquake thatis expectedwithin

the nationalterritoryhas a surface -wave magnitude Ms notgreaterthan 5, then itis

recommendedthatthe Type2 spectrum shouldbe adopted..

"The elastic response spectrum Se(T)forthe reference return, periodis definedbythe

following expressions:


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design spectrum parameters, EC8


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EC8 5% damped, elastic spectra. Type 1 spectraare enrichedin long periodandare suggested

forhigh seismicityregions.Conversely, Type2 spectraare proposedforlowto moderateseismicityareas (like France), andexhibitbothalargeramplification atshortperiod, anda

much smallerlong periodcontents, withrespectto Type 1 spectra. These propositions,

however, were constrainedusingfew events mostlyrecordedon analogicalinstruments.

the parameters S, TB, TCandTDaregiven in tablesforthe differentsubsoiltypesforeachof

the twotypes ofspectra.


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The influence of near-surface geology on response spectra is taken into account via the soil

parameter S. The fact that locations underlain by soil deposits generally experience

stronger shaking than rock sites during earthquakes has been recognised for many years,

both from field studies of earthquake effects and from recordings of ground motions. The

influence of surface geology on ground motions is now routinely included in predictive

equations. Code specifications of spectral shapes for different site classes generally reflect

the amplifying effect of softer soil layers, resulting in increased spectral ordinates for such

sites, and the effect on the frequency content, which leads to a wider constant accelerationplateau and higher ordinates at intermediate and long response periods.

EC8 is unique amongst seismic design codes in that it is actually a template for a code

rather than a complete set of definitions of earthquake actions for engineering design. Each

member state of the European Union will have to produce its own National Application

Document, including a seismic hazard map showing PGA values for the 475-year return

period, select either the Type 1 or Type 2 spectrum and, if considered appropriate, adapt

details of the specification of site classes and spectral parameters.

The basic mechanism for defining the horizontal elastic design spectrum in EC8 is outdated

and significantly behind innovations in recent codes from other parts of the world, most

notably the US. It is to be hoped that the first major revision of EC8, which should be carried

out 5 years after its initial introduction, will modify the spectral construction technique,

incorporating at least one more anchoring parameter in addition to PGA.

In the Luso-Iberian peninsula, seismic hazard is the result of moderate-magnitude local

earthquakes and large-magnitude earthquakes offshore in the Atlantic. The Spanish seismic

code handles their relative influence by anchoring the response spectrum to PGA but then

introducing a second set of contours, of a factor called the contribution coefficient, K, that

controls the relative amplitude of the longer-period spectral ordinates high values ofK

occur to the west, reflecting the stronger influence of the large offshore events.

5.2 NCSR-02 (Spanish Seismic Design Code) Elastic Response Spectrum.


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parametersfordetermining elastic response spectrum, NCSR-02.


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elastic response spectrum, NCSR-02.

Are We Ready for El Centro?

I shall firstly construct the design spectrum for Malaga City, according to NCSR-02.

According to the zonation map (hazard map) (see NCSR-02), we have that the values of ab

and K are respectively 0.11g and 1. Assuming a construction of normal importance, we take

the importance factor equal to unity ( = 1). Also, assuming the site is located on firm soil

(hard rock), C = 1, and the soil coefficient is found to be S = 0.8067 .


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Elastic Design SpectrumforMalagaaccordingtoNCSR -02.

Comparison of El Centro Design Spectrum with NCSR-02 Design Code for Malaga

I shall plot this design spectrum in MATLAB, as well as the Newmark-Hall pseudo-

acceleration design spectrum for El Centro, for a 50% prob. of exceedance, for systems with

a 5% damping ratio. The NCSR spectrum corresponds to a 10% of exceedance in a period of

50 years (return period 475 yrs). The code for generating the plot can be found in the m-

file: ComparisonNewmarkNCSR02.m.


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Comparison ofNewmarkDesign SpectrumforElCentro(50% probexceedance) with

Elastic Design SpectrumforMalagagiven byNCSR -02,forstiffsoilandareturn periodof475

yrs.

References.

Anil K. Chopra "Dynamics ofStructures".

FEMA notes

FarzadNaeimThe Seismic Design Handbook.

Response OfStructures toEarthquake GroundMotions .

Lecture 18 -Earthquake Response Spectra..

Ahmed Y. Elghazouli (edited by), Seismic Design ofBuildings toEurocode 8

European Seismological Commission.UNESCO-IUGS International Geological Correlation

Program Project no. 382 SESAME

Eurocode 8

NCSR-02

El Centro Accelerogram Data

(individual files I used for reading data to MATLAB):

y Acceleration Datay Velocity Data


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y Displacement Data

Documents

Are We Ready for ELcentro