yrc 2010 sfee ionut damian articol 1

7/30/2019 YRC 2010 SFEE Ionut Damian Articol 1

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SPECTRUM COMPATIBLE ACCELEROGRAMS: METHODS OF

GENERATION AND PARAMETRIC STUDIES

IONUT DAMIAN - Ph.D. student, Technical University of Civil Engineering Bucharest,

e-mail: [email protected]

Rezumat: n ultimul timp analiza dinamic este din ce n ce mai utilizat pentru proiectareastructurilor civile. Cu toate acestea, acceleraia seismic considerat ca dat de intrare estecea mai mare necunoscut a acestui tip de analiz. n Romnia exist numai cteva nregistrricare pot fi utilizate pentru acest tip de analizi de aceea accelerogramele artificiale suntntrebuinate foarte des. Exist mai multe tipuri de accelerograme artificiale, n funcie de

parametrii asemntori cu cei ai unui seism natural. Practica uzaual, care este aplicati ncodurile de proiectare, este de a genera accelerograme sintetice compatibile cu spectrul elasticde acceleraii specific amplasamentului. Exist dou metode de generare a acestui tip deaccelerograme: prima este bazat pe teoria vibraiilor aleatoare, iar cea de-a doua pornete dela o accelerogram natural care este modificat iterativ pn cnd spectrul acesteia devine

spectrul int. Articolul prezint principiul fiecrei metode i analizeaz influena fiecruiparametru de intrare asupra accelerogramelor generate. n final sunt comparate cerinele deductilitate ale acelerogramelor generate cu cele dou metode.

Abstract: Lately, dynamic analysis became more and more utilized for the design of civilstructures. However, ground motion, which is considered as input, is the main unknown of thisanalysis. In Romania, there are only a few recordings that can be used for this kind of analysis,and therefore artificial accelerograms have to be used. There are several types of artificialground motions, depending on the parameters that resemble the ones from the naturalrecordings. The common practice, which is also applied in design codes, is to generatesynthetic accelerograms that are compatible with the elastic acceleration spectra of the site.There are two methods to generate this kind of ground motions: the first one is based onrandom vibration theory, and the second one starts from a natural motion and iteratively

changes it until the spectra of the modified accelerogram resembles the target spectra. Thepaper presents the principle of each method and studies the influence of each input parameteron the generated ground motions. Finally, the ductility demands of accelerograms generatedbased on the two methods are compared.

Keywords: spectrum compatible accelerograms, random vibration theory, modification of

initial accelerogram, ductility demand

1. General Consideration

Dynamic analysis has been intensively used in the last years for the evaluation of existing buildings or

as a design procedure. This kind of analysis has a lot of unknown parameters in the structural model,such as mass and loadings, stiffness (in the case of reinforced concrete structures), and damping for

linear analyses or the hysteresis behaviour of the members in nonlinear analyses. However, the main

unknown is the input acceleration history due to the random character of earthquakes.

In modern design codes, for dynamic analysis, input acceleration can be modeled by artificial

accelerograms that are compatible with the elastic acceleration spectra of the site. Romanian code for

seismic design, P100/2006 specifies that (1) a minimum of three accelerograms should be used, (2) the

mean of Peak Ground Acceleration (PGA) of the accelerograms to be higher than the design ground

acceleration of the site, and (3) the mean of their spectral acceleration to reach a minimum of 90% of

the elastic acceleration spectra of the site for all periods.

Computation of artificial accelerograms has always been a general topic for structural engineers. The

first impulse was to generate accelerograms whose elastic spectrum resembles the elastic spectrum ofthe site, maybe because this is somehow an insightful method. A quite similar method is to generate

accelerograms based on previously defined parametric functions of Power Spectral Density (PSD),


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such as Kanai-Tajimi or Clough-Penzien. The most refined methods are based on the attenuation

function of intensity measure (PGA or spectral acceleration) and the geotechnical characteristics of the

site. However, this article will focus only on spectrum compatible accelerograms. For this study the

target spectrum is that from the Vrancea seismic zone, characterized by a long corner period of 1.6s.

There are two principal methods that can be used to compute artificial accelerograms to mach a target

elastic acceleration spectrum. One is based on Random Vibration Theory (RVTM) [1], the other either

starts from a real accelerogram or other signal and then iteratively changes it until the spectrum of theaccelerogram resembles the target spectrum [2]. This will be called Modification of Initial

Accelerogram Method (MIAM). Although the accelerograms based on both forementioned methods

have the same elastic spectrum, it is interesting to study if the ductility demand is the same in the case

of nonlinear behaviour. This study will be performed on Single Degree of Freedom (SDOF) systems.

2. Presentation of the Methods and Sensitivity Study

2.1. Modification of initial accelerogram method

This is the most intuitive method. As it is known, any signal can be decomposed in harmonic series.Each period is characterized by an amplitude and a phase angle. The basic assumption of the method is

that the spectral response for a particular period depends only on the Fourier amplitude of excitation

for that particular period. This is due to the fact that in the frequency domain, the transfer function

between the excitation and the response has a large peak at the predominant period of the system and

negligible values elsewhere, making unimportant the harmonics of the input with other periods.

However, as it can be inferred from figure 1 (left), the transfer function has finite values for systems

that are provided with a minimum amount of Percentage of Critical Damping (PCD). For 0 PCD the

transfer function has infinite values at the predominant period of the system, so the method will fail in

this case.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

2 4 6 8 10 12

w[rad/s]

|H(iw)| x= 0 . 05

x= 0 . 02

x= 0 . 00 5

0

1

2

3

4

5

6

78

9

0 1 2 3 4 5

T[s]

Sel[m/s2]

Sa target

Sa iteration i

Sat(Tk)

Sai(Tk)

Tk

Fig.1 - Transfer function for Tn=1s and several percentages of critical damping (left) and computation of the

amplification factor from iteration i to iteration i+1 (right)

The steps to generate a new accelerogram (within one iteration) are:

i) Compute the spectrum of the accelerogram

ii) Compute Fourier transform of the accelerogram

iii) For each period, amplify the Fourier transform of the accelerogram with the ratio between target

spectrum and actual elastic spectrum computed in step i) (figure 1, right).

The iterations are carried on until a declared maximum number of iterations are reached. The

accelerograms computed by this method maintain some characteristics of the initial signal. Because

each iteration the whole Fourier transform of one period is multiplied by a real number, the phase

angles of the artificial accelerogram are the same as the phase angles of the initial accelerogram.

Another attribute of the generated accelerograms is that they maintain the same strong motion duration

as the starting accelerogram. Figure 2, left presents the initial and the generated accelerogramcompatible with the Vrancea spectrum, with Tc=1.6s and ag=0.20g. The initial accelerogram is the


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recording from Takatori station, Kobe earthquake. PCD was set to 5% and a number of 25 iterations

have been used.

-6

-4

-2

0

2

4

6

0 10 20 30 40

t[s]

ag[m/s2]

Initial

Generated

0.0

0.1

0.20.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0

t/Tr

NCE Initial

Generated

Fig. 2 -Temporal signals (left) and normalized cumulative energy (right)

In any iterative strategy the iterations are carried on until the error is less than an acceptable error. In

the case of synthetic accelerograms, the convergence needs to be checked for all periods considered,

making the convergence very difficult or almost impossible to attain. Therefore, a maximum number

of iterations should be declared, and when this number is reached, the process will stop.

Fig. 3 - The influence of maximum number of iterations

Figure 3, left presents the influence of the number of iterations for the same Kobe accelerogram. It can

be inferred that from ten iterations the efficiency per iteration decreases. This can be observed fromthe right part of the figure which presents the convergence of the spectrum to target spectrum for a

system with 1s period.

The length of the accelerogram is another parameter that influences the quality of spectrum matching.

The matching quality seems to decrease with the length of the recording. This is due to the fact that

longer recordings approach a steady-state condition, and because the discrete Fourier transform is

better defined. Figure 4 presents matching comparison between two accelerograms with different

recording lengths: one is the accelerogram from the 10th

of May, 1986, recorded on Expozitiei

Boulevard with 14s length, and the other is the North-South Incerc recording from Vrancea 1977 that

has 45s. PCD was set to 5% and a number of 25 iterations were used.

0

1

2

3

4

5

6

0 1 2 3 4 5

T[s]

Sel[m/s2]

Target

Tr=14s

Tr=45s

Fig. 4 - The influence of recording length

0

1

2

3

4

5

6

7

0 1 2 3 4 5

T[s]

Sel[m/s2]

Target

5 ite

10 ite

15 ite

20 ite

25 ite

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20 25

No. iterations

Sai(T=1s)/Sat(T=1s)


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As it was previously demonstrated theoretically, MIAM fails to compute accelerograms that resembles

the target spectrum for 0 PCD. However, this will be numerically investigated next. In the presented

example 25 iterations were used and the starting accelerogram is one of the recordings from the Kern

County earthquake, Taft, which is 55s long.

Fig. 5 - The influence of PCD (left) and Normalized Autocorrelation Function (NAF) (right)

From the left plot of figure 5, a band of periods can be observed where spectrum of the generated

accelerogram changes values fundamentally. The bandwidth is 0.8s and has the center in the vicinityof the corner period of the spectrum. From the study of the normalized autocorrelation function (figure

5, right) it can be seen that the predominant period of the accelerogram has been increased, and now

we are talking about a wide band process, with its center period around the corner period and 0.8s

width. So far, it is known that for small PCD, the mechanical admittance approaches the infinite in the

vicinity of the fundamental period of the system. However, for periods near to the corner period, the

discrete Fourier transform of the signal has also maximum values. Therefore, changes that are

relatively small in input signal produce significant changes in output signal, and, as a result, a close

spectrum matching will never be reached.

2.2. Random vibration theory method

The method was first implemented in SIMQKE software [1]. Its basic idea is that any signal can be

decomposed in a sum of harmonics.

1

)sin()(n

kkkkg tAtu

=

+= && (1)

In formula (1), kA represents the amplitude of kth

harmonic, k the circular frequency of the same

harmonic, and k the phase angle. One can propose the total time of the recording and the time

increment of the accelerogram, which result in the circular frequencies of the harmonics. The next step

is to generate random phase angles between 0 and 2. So far, the only unknown in formula (1),

remains the amplitude of the sinusoids. One can consider only positive harmonics, and the total power

of the signal )t(ug&&

equals half of the sum of squares of the amplitude of all sinusoids.

=

=

n

1k

2k

2

AP (2)

In the case of uniform generated circular frequencies, a new function can be computed in frequency

domain, so as2

)(2

kk

AS = , where

rT

2 = is the difference between two subsequent circular

frequencies. The new function, )(S , is the PSD of the signal and relays the contribution of each

harmonic in the total power. In Random Vibration Theory (RVT), the value of the output that has pprobability not to be exceeded in a signal that lasts Tr, is expressed as a multiple of standard deviationof the output:

)(,, rxpTrpTr Trx = (3)

0

2

4

6

8

10

12

14

0 1 2 3 4 5

T[s]

Sel[m/s2]

Target

Generated

BT=0.8s

Tc=1.6s

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

D[s]

NAF Initial

Generated

BT=0.8s

Tc=1.6sTp=0.7s


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In equation (3), p,Trr is the peak factor associated with duration Tr andp nonexceedence probability,

while )T( rx is the standard deviation of the displacement associated with the same duration Tr. The

PSD of input signal )(S and the PSD of output signal )(Sx are connected through2

)i(H which

is the mechanic admittance of the system.

)()()(

2

SiHSx=

(4)The dispersion of the output equals the area under the PSD. It was demonstrated that the dispersion in

steady-state conditions can be decomposed in a resonant part and in a nonresonant part, due to the

forementioned observation that the transfer function has significant values only if adjacent to the

fundamental period of the system.

])(1

[1)]-4

()(

[

)()()(

043

2,

2,

2

0

0

2

dSS

dSiHdS

n

nn

n

nerezxrezxxx

+=

=+===

(5)

One can observe that when PCD 0 or when the natural circular frequency 0n the dispersionaccedes infinite. It was shown that the value of the dispersion increases from 0 to its maximum value

at the end of the excitation, revealing its transient process (Caughey & Stumpf, 1961). This is why

PSD depending on excitation length was introduced (Corotis & Vanmarke, 1975). The great advantage

of this approach is that the singularity is eliminated when 0 . In this case the resonant dispersionhas the following expression:

)(-)e-(1

4

)()(

3

T2-

3

2,

rn

n

n

n

nrrezx

SST

= (6)

The determination of peak factor p,Trr is a very difficult topic that involves not only analytical

solutions, but deterministic parameters as well. To determine p,Trr , the kth

spectral moment of the PSD

of output and two important parameters that define the frequency content of the output signal shouldbe mentioned here:

xx

xrx

x

xrx

rxk

rxk

T

T

dTST

,2,0

2,1

,0

,2

0

,

-1)(

)(

),()(

=

=

=

(7)

It can be seen that )T( rx represents a type of center mass of PSD and that )T( rx measures the

dispersion of the circular frequency of the output relative to )T( rx . For a narrow band process

)T( rx will accede 0, while for a wide band process it will accede 0.5. This value is reached for white

noise.

The most important step in finding the peak factor is the average number of times per second the

response will exceed a value ax = (Rice, 1945):

2

2

2

1-

2

x

a

xa e

= (8)

Theoretical and experimental studies have shown that the probability that a random variable, x, not toexceed a certain threshold, ax = , in the time of the accelerogram has the following form:


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rTr eATp

-)( = (9)

In the previous formula, A represents the probability that the starting value of the random variable tobe less than ax = and its value is generally 1. represents a decreasing parameter. Crandall (1966)proposes the next formula for this parameter:

2

2

2

1-

2

-

-1

-12

x

xe

a

a

a

e

e

= (10)

2.1xe = is an empirical frequency band indicator. By combining the last three relations and

applying certain approximations, results in the peak factor expression in a steady-state process:

1-

5.0)2ln(-,

)]ln(-[2

)]}-1(2ln[2{

pT

n

enr

rx

npTr

e

=

=

(11)

In the case of transient processes, Vanmarke (1976) proposes the next formula for the nonexcedence

probability of a certain threshold in a certain time, Tr:

0rTr e)T(p

-= (12)

0rT is the equivalent steady-state length and is calculated with the formula:

rn

rn

T

T

mrr

e

em

eTT

-

2-

)1-(2-0

-1

-1

=

(13)

For short periods, elastic acceleration spectrum will reach PGA and thus frequency content indicators

of the output signal resembles the ones of the input signal. For medium and long periods Vanmarke

uses the following expressions for the frequency content indicators:

trx

nrx

T

T

4)(

)(

=

=

(14)

After generating the PSD of the signal, the amplitude of each harmonic is calculated and the

accelerogram is computed based on equation (1). However, to provide the accelerogram with a natural

aspect, an intensity envelope should be applied. This is as if applying a filter in the time domain. The

envelope is deterministic and different models have been proposed: trapezoidal (Hou, 1968),

exponential (Shinozuka, 1973), compound (Jennings, 1968). Additionally, iterations can be carried on

the generated signal, if desired, as in MIAM.

The steps to compute a new signal are:i) Propose the number of points of the accelerogram and the time increment

ii) Calculate circular frequencies and randomly generate phase angles for each harmonic

iii) Compute PSD of the input motion based on peak factor and target spectrum

iv) Compute the amplitude of each harmonic

v) Compute the input accelerogram based on relation (1)

vi) Apply time envelope, adjust peak acceleration if applicable and apply base line correction


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

t/Tr

I(t/Tr)

Trapezoidal

Exponential

CompoundI(t)=I0(e

-t-e

-t)

I(t)=(t/tp)n

tp/Tr= tc/Tr

Fig. 6. - Models of intensity envelope

In addition to the parameters discussed at MIAM, the influence of the phase angle could be studied.

The left plot of figure 7 presents the spectra generated for three synthetic accelerograms that have the

same amplitudes but different phase angles. The trapezoidal envelope has been applied to all signals. It

can be observed that the phase angle can be very important for certain periods. This happens because it

could be possible that two close periods have the same phase angle, resulting in a peak for that period.Or the two phase angles could be phase shifted with radians and annihilate each other.

Fig.7 - Phase angle influence (left) and envelope influence (right)

Because the envelope is a filter applied in the time domain, it can change the spectrum of the

generated signal. This can be seen in figure 7, right, where for the same generated signal different

envelopes were applied. Excluding the exponential envelope, the rest of the envelopes seem not to

influence the spectrum. However, the exponential envelope is a special case that can generate pulse

type accelerograms and whose parameters need to be iteratively calibrated.

To study the efficiency of time dependent transfer function, the influence of PCD and length of

recording will be studied again. Figure 8, left presents the spectrum of three acccelerograms that have

the same record length, 40.96s, and the same phase angles, but different PCD. As it can be seen, atnull PCD there are the same problems as at MIAM. However, for short records the stationary problem

seems to be solved, as it can be observed in the right plot of figure 8.

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5

T[s]

Sel[m/s2] Target

=5%

=2%

=0%

0

1

2

3

4

5

6

7

0 1 2 3 4 5

T[s]

Sel[m/s2] Target

Tr=10.24s

Tr=20.48s

Tr=40.96s

Fig. 8 - PCD influence (left) and recording length influence (right)

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5

T[s]

Sel[m/s2] Target

Phase 1

Phase 2

Phase 3

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5

T[s]

Sel[m/s2]

Target

No envelope

Trapezoidal

Exponential

Compound


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3. Ductility Demand of Generated Accelerograms

In this section aggressiveness will be measured in the ductility demand of SDOF systems. Of course,

there are a lot of damage indices that can be used, such as the Park-Ang, the models based on period

elongation, and others. However, in the case of reinforced concrete structures, the tests highlights that

the most important damage parameter is the ductility demand.

Fig. 9 - Mean ductility demand (left) and COV (right)

0

1

2

3

4

5

6

7

8

910

0.0 1.0 2.0 3.0 4.0

T[s]

mm MIAM

RVTM

cy=0.10

0

0.02

0.04

0.06

0.08

0.1

0.12

0.140.16

0.180.2

0.0 1.0 2.0 3.0 4.0

T[s]

COVm

MIAM

RVTMcy=0.10

0

1

23

456

7

89

10

0.0 1.0 2.0 3.0 4.0

T[s]

mm MIAM

RVTMcy=0.15

0

0.02

0.04

0.06

0.08

0.10.12

0.14

0.16

0.18

0.2

0.0 1.0 2.0 3.0 4.0

T[s]

COVm

MIAMRVTM

cy=0.15

0

12

3

4

5

67

8

9

10

0.0 1.0 2.0 3.0 4.0

T[s]

mm MIAM

RVTMcy=0.20

0

0.05

0.1

0.15

0.2

0.25

0.0 1.0 2.0 3.0 4.0

T[s]

COVm

MIAM

RVTM

cy=0.20

0

1

2

3

4

5

6

0.0 1.0 2.0 3.0 4.0

T[s]

mm MIAM

RVTM

cy=0.25

0

0.05

0.1

0.15

0.2

0.25

0.0 1.0 2.0 3.0 4.0

T[s]

COVm

MIAM

RVTM

cy=0.25

0

1

2

3

4

5

0.0 1.0 2.0 3.0 4.0

T[s]

mm MIAM

RVTM

cy=0.30

0

0.05

0.1

0.15

0.2

0.25

0.0 1.0 2.0 3.0 4.0

T[s]

COVm

MIAM

RVTM

cy=0.30


9/10

As it can be inferred, the two generation methods are quite similar. However, in MIAM, the artificial

accelerogram preserves some of the characteristics of the initial accelerogram, such as phase angle and

strong motion duration. In this case it is interesting to test if the accelerograms generated with the two

methods have the same ductility demands. For this study two sets of 21 spectrum compatible

accelerograms will be used: one set generated with MIAM and the other generated with RVTM. The

target spectrum is the one of Vrancea seismic zone, with a corner period of 1.6s and a value of 0.24g

for design ground acceleration. Ductility spectra will be computed for 5 strength levels of normalized

yielding force: 0.1, 0.15, 0.2, 0.25 and 0.3. The hysteretic model is a bilinear Takeda, which is thesimplest model that takes into account the complex behaviour of reinforced concrete elements. The

postelastic slope is 5% of elastic slope. The unloading stiffness will be reduced depending on the

maximum ductility reached. The value of the unloading parameter is 0.3, a medium value, and PCD

was considered 5%.

Figure 9 presents mean ductility demand (left) and Coefficient of Variation (COV) of ductility demand

(right). It seems that there is no significant difference on median ductility between the two sets of

accelerograms. Likewise, the same thing can be said about the COV.

The significant differences between the two ductility demands were reported only for large values of

normalized yielding force. However, they have been reported for stiff periods less than 0.5s where

structures are provided with significant overstrength factors.

4. Conclusions

The article presents two methods to compute spectrum compatible accelerograms. One is based on a

procedure that iteratively changes an initial accelerogram (MIAM), while the other is based on random

vibration theory (RVTM).

It was revealed that in MIAM the matching quality decreases with the decrease of the record length.

This happens because when recording length is long the process approaches a steady state, whereas

when recording length is short it approaches a transient state. However, due to the use of time

dependent transfer functions, this drawback is not signalized by RVTM.

Furthermore, the matching quality is influenced by the percentage of critical damping. All the methods

failed to compute accelerograms whose spectra resemble reasonably the target spectrum in the vicinity

of corner period at 0 percentage of critical damping. The explanation is that the accelerograms

generated are wide band processes with the central period near the corner period. Therefore, in that

region both the transfer function and the Fourier transform of excitation have large values. So, a small

change in input signal will produce a large change in output signal, making matching impossible.

The optimum number of iterations is 10. For greater values, the efficiency per iteration decreases

significantly.

The phase angle is very important for the initial signal. However, its importance reduces to 0 when

iterations are carried on to compute artificial accelerograms.

Finally, although the characteristics of generated accelerograms by the two methods are quitedifferent, no significant difference has been reported in ductility demand and COV. For the spectral

acceleration corresponding to a design ground acceleration of 0.24g, the mean COV reported has a

value of 0.15, which is quite a small value.

References

[1] Gasparini D.A., Vanmarke E. H. - Simulated Earthquake Motions Compatible with Prescribed ResponseSpectra. Rpt. No. R76-4, Dept. of Civil Engrg., MIT, Cambridge, Massachusetts, 1976

[2] Naumoski N.D., - Program SYNTH. Generation of Spectrum Compatible Accelerograms Compatible with aTarget Spectrum, 1988

[3] Shome, N., Cornell, C.A. - Normalization and Scaling Accelerograms for Nonlinear Structural Analysis. 6thUS National Conference on Earthquake Engineering, 1998

[4] Vacareanu, R., Aldea, A., Lungu, D. - Structural Reliability and Risk Analysis, Conspress, Bucharest, 2007


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[5] EN 1998-1 -Design of Structures for Earthquake Resistance Part 1: General Rules, Seismic Actions andRulles for Buildings, 2004

[6] P100-1 - Seismic Design Code Part 1: Seismic Design of Structures, 2006

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