Sdee Response Spectrum Theory

SDEE04-RESPONSE SPECTRUM THEORY 09/10/2015

Prof. R.S. Desai 1

Presented byProf. R. S. DESAI

STRUCTURAL DYNAMICS &

EARTHQUAKE ENGINEERING

WELCOME TO YOU ALL09/10/2015 1RESPONSE SPECTRUM

STRUCTURAL DYNAMICS &

EARTHQUAKE ENGINEERING

UNIT-IVRESPONSE SPECTRUM THEORY

09/10/2015 2RESPONSE SPECTRUM

Get a set of several SDOF structures each having its own period of vibration, such that all have periods uniformly increasing from a minimum period T1 to a maximum specified value of PERIOD Tn. (T1,T2,T3,………Tn)

The interval (dT) between any two consecutive periods is constant.

Consider a single STRONG GROUND MOTION (i.e an ground motion accelerogram)

RESPONSE SPECRUM


0 5 10 15 20 25 30 35-400

-300

-200

-100

0

100

200

300Chamoli, Comp-N20E

TIME (sec) A

ccel

erat

ion

(cm

/sec

2 )


0 2 4 6 8 10 12 14-500

-400

-300

-200

-100

0

100

200

300

400

500Koyna Dam, Comp-Longi.

TIME (sec)

Acc

eler

atio

n (c

m/s

ec2 )


Prof. R.S. Desai 2

Each of the above SDOF structure is subjected to the selected ground motion excitation at BASE for full duration of the accelerogram.

The PEAK RESPONSE (displacement, velocity, acceleration ) of each SDOF structure is noted and plotted on Y axis corresponding to its PERIOD on X axis.

Resulting graph of the PEAK RESPONSE on Y v/s PERIOD on X represents a RESPONSE SPECTRUM.

RESPONSE SPECRUM

09/10/2015 5RESPONSE SPECTRUM 09/10/2015 6RESPONSE SPECTRUM

09/10/2015 7RESPONSE SPECTRUM 09/10/2015 8RESPONSE SPECTRUM


Prof. R.S. Desai 3

All three spectra are skillfully used in defining or constructing the designresponse spectrum, which will be discussed later.

Therefore, a combined plot showing all three of the spectral quantities isdesirable. This integrated presentation is possible because of the relationshipthat exists between these three quantities.

log Sd =log Sv - log ωnlog Sa = log Sv +log ωn

From Equations , it is evident that a plot on logarithmic graph paper with logSv as ordinate and log ωn as abscissa, the two equations are straight lines with slopes -45 and +45 for constant values of logSd and logSa, respectively.

If log of time period T, instead of log ωn, is considered, then these orthogonal straight linesare interchanged. Thus, a four way log plot as shown in Figure can be used to plot all three spectra

09/10/2015 9RESPONSE SPECTRUM Idealized response spectrum by a series of straight lines for El Centro earthquake09/10/2015 10RESPONSE SPECTRUM

End of Response Spectrum Theory and its Construction.


Design Response Spectrum and its Construction



Prof. R.S. Desai 4

DESIGN RESPONSE SPECTRUM is intended to be used for the design ofnew structures or the risk evaluation of existing structures for futureearthquakes, which are not known.

a. The spectrum should be as smooth as possible and devoid of high irregularities, as observed inthe response spectrum of an earthquake shown in Figure , for two reasons. Firstly, irregularitiesin the spectra of two different earthquakes could be significantly different, leading to anerroneous estimate of the spectral ordinates for future earthquakes. Secondly, for a highlyirregular spectrum, spectral ordinates may drastically change for a small change in frequency.As there is always a certain amount of uncertainty in the determination of natural frequencies ofstructural systems, an irregular spectrum can provide a very erroneous estimate of theearthquake forces calculated for the structures. Therefore, a design spectrum in a four way logplot is expected to consist of a series of straight lines, such as that of an idealized spectrum shownin Figure

Requirements to be Satisfied by DRS: (CRITERIA of CONSTRUCTION)


b. The design spectrum should be representative of spectra for past earthquake ground motions in theregion. If there are insufficient or no earthquake records available for the region, then records of othersites under similar conditions may be used. The factors that should be considered for identifyingsimilar conditions include the magnitude of the earthquake, distance of the site from the fault, faultmechanism, geology of the travel path of the seismic waves from the source to the site, and the local soilconditions of the site.

c. A single response spectrum may not be able to represent the variations in the spectra of all pastearthquakes in the region. Therefore, two response spectra, one mean spectrum, with the other beingthe mean plus one standard deviation spectrum, should be considered as design spectra.

d. The design response spectrum should be a normalized response spectrum with respect to the peakground acceleration (PGA), as thePGAmay drastically vary from one place to another. Furthermore, adesign response spectrum should be consistent with the specification of the level of the seismic designforce, or the deformation of structures experienced during previous earthquakes.

Finally, the design response spectrum should be consistent with seismic design philosophy. Currently,a dual design philosophyis adopted (which will be discussed later). This requires specification of two


Design Response Spectrum -Construction


1. Expected PGAvalues for design and maximum probable earthquakes are derived for the region using the procedure of hazard analysis.

2. Peak values of the ground velocity and displacement are estimated using empirical relationships valid for the region, which are given in the form of

ůgmax=c1* ügmax/g ugmax=c2* ůgmax/ ügmax

3. The values of c1 and c2 are determined from the recorded earthquake data. Typical values of c1 and c2 may be taken as c1 =1:22 to 0:92 m.s-1 and c2=6.



Prof. R.S. Desai 5

4. On the four way log graph paper, plot the baseline showing ügmax, ug maxand ůgmax as hown in Figure. Multiply These quantities by theamplification factors αA, αD, and αV, respectively, to obtain the lines bc, de,and cd.

b

cd

e

fa


Point b corresponds to frequency f2=1/T2, which may be taken as f2=4f1 wheref1=1/T1, is the frequency corresponding to the intersection point. Note thatintersection points c and d are fixed by the relative values of αA, αD, and αV.Point a corresponds to the frequencyf3=1/T3, which may be taken as f3=10f1.Points e and f are selected corresponding to very lowfrequencies (large periods),and could be of the order of (1/10–1/15) and (1/30–1/35) Hz,respectively.

b

c d

e

fa


Exact values of time periods corresponding to points a, b, e, and f depend upon recorded data in theregion. Similarly, values of aA, aD, and aV also depend upon the recorded earthquake data. Inreference [4] some representative values of aA, aD, and aV for mean and mean plus one standarddeviation spectra obtained from a set of large earthquake data are given. Note that the values of aA,aD, and aV depend upon the damping, as expected.

Once the design spectrum is drawn in a four way log plot, the normalized acceleration responsespectrum can be obtained in an ordinary plot. A typical plot of the normalized pseudo accelerationspectrum (derived from a log plot) as given in the codes of practice is shown in Figure below. It isseen from the figure that spectral accelerations (Sa) for a soft soil profile are more compared withthose of a hard soil profile at periods of more than 0.5 s. This is the case because the amplifiedfactors aA, aD, and aV substantially change with the soil conditions.

09/10/2015 19RESPONSE SPECTRUMDesign Acceleration Response spectrum as given in IS1893 Code



Prof. R.S. Desai 6

End of D.R.S.and its Construction


STRONG GROUND MOTION&

It’s 3 Characteristics

0 5 10 15 20 25 30 35-400

-300

-200

-100

0

100

200


TIME (sec)

Acc

eler

atio

n (c

m/s

ec2 )

“Accelerograph” is an instrument used to obtain a graphical measurement of an Earthquake ground shaking acceleration called “accelerogram” as shown above.09/10/2015 22RESPONSE SPECTRUM

STRONG GROUND MOTIONis a ground motion accelerogram orportion of it which is capable ofaffecting life on earth or itsenvironment.(by Krammer)

The amount of energy arriving at a surface dependson so many parameters which is estimated byEarthquake Hazard Analysis of a region. In generalacceleration of magnitude more than 0.2g or Anearthquake mag. Of more than 5 on Richter scalecould be considered as S.G.M.


Important characteristic of ground motion of engineering significance

i) Amplitudeii) Frequency contentiii) Duration

Several parameters are measured/calculated to extract above characteristics from a given accelerogram , and used for Earthquake Analysis and Design purposes. These are given below.

0 5 10 15 20 25 30 35-400

-300

-200

-100

0

100

200


TIME (sec)

Acc

eler

atio

n (c

m/s

ec2 )



Prof. R.S. Desai 7

A. PARAMETERS for AMPLITUDE


Parameter Definition, merits, & properties of parameters1. Motion parameters

from time history ofground motion area.accelerationb.velocityc.displacement

Generally one of the quantity is directly obtained by instrumentalmeasurement and others obtained by integration/ differentiationvelocity obtained by integration of acceleration filters out higherfrequencies. Displacement are further more smoothed.

2. P.H.APeak horizontal acceleration

Largest magnitude (absolute value) of the time history of a singlecomponent (X or Y). Resultant P.H.A is obtained by vector sum of X &Y components. This is highly useful parameter as it is directly related toinertial forces developed in structure. The correlations of P.H.A &INTENSITY of earthquake are useful where magnitude of earthquakeis not available. Vertical acceleration has no significant effects onstructure.

P.H.A is important parameter related to damage but not alwaysgoverning. It lacks information regarding frequency content &duration very high frequency P.H.A will not cause damage than whatmore cycles of single weaker acceleration do.

3. Peak velocity &peak displacement

For structures or facilities sensitive to intermediate-frequency range(tall building & bridges) these parameters may be more accurateindication of potential damage than P.H.A

Peak displacement is subjected to processing and other errors init, is rarely used in analysis. Peak displacement is related to lower-frequency components of g.m.


B. PARAMETERS FOR FREQUENCY CONTENT


Parameter(groundmotionspectra)

Definition, merits, & properties of parameters

1. Fourierspectra

A plot of Fourier amplitude versus frequenncy is called Fourieramplitude spectrum and plot of phase angle with respect tofrequency gives Fourier phase spectrum. From plots indicationof frequency content is easily obtained.

Fourier acceleration amplitudes tend to be largest over anintermediate range of frequencies bounded by lower frequencycalled (corner frequency) and cutoff frequency. It is inverselyproportional to cube root of seismic moment.(Brune 1970,71)



Prof. R.S. Desai 8

3. Responsespectra

When axis is time period spectral acceleration andspectral displacement reverses their places.At low frequency spectral displacement is nearlyconstant & at high frequency spectral acceleration isnearly constant & at intermediate frequency spectralvelocity is nearly constant.

So very often displacement spectra is divided intothree regions acceleration controlled (hi frequency),velocity controlled (intermediate frequency) anddisplacement controlled (low frequency).


C. PARAMETERS FOR DURATION


Duration of S.G.M has importance in number of reversalcycles of motion. “Duration is related to the time required forrelease of accumulated strain energy by rupture along fault.”Since strain energy release is related to fault length, rupturearea: (Hanks & Mc Guire 1981) suggested that durationshould be proportional to

a. Bracketed duration of S.G.M is more commonly used (bolt 1969) timespan between first and last exceedance of threshold accn. (Usually 0.05g)

b. Time span between 5% to 95% of total energy is recorded (Trifunac &Brady 1975).

c. Corner period (inverse of corner frequency) (Boore 1983).

d. Power spectral density concepts of duration are also proposed.09/10/2015 31RESPONSE SPECTRUM 09/10/2015 32RESPONSE SPECTRUM

Documents

Sdee Response Spectrum Theory