**Proceedings of the 20th International
Conference on Nuclear Engineering**

**ICONE20**

**July 30 - August 3, 2012, Anaheim, California, USA**

ICONE20POWER2012-54187

Generation of Response Spectra Compatible Artificial Acceleration Time Histories

Fortum Power and Heat Ltd. Espoo, Finland |
Pentti Varpasuo Fortum Power and Heat Ltd. Espoo, Finland |

Abstract

A procedure to generate artificial acceleration time histories compatible with predefined target response spectrum is presented. The procedure is demonstrated with three examples.

It was found out that the generated artificial histories yield high quality response spectra for single-damping defined target spectra. A need for method that yields histories matching target spectra defined with multiple damping values was recognized.

The response spectrum method is not always adequate from the designer's point of view. The problem at hand may be nonlinear or the very approximate nature of the response spectrum method is questioned. Therefore, time history analysis must be performed. In order to perform analysis, the designer needs response spectrum compatible acceleration time histories.

Various methods of generation of artificial (i.e. "synthetic") time histories compatible with target response spectra have been proposed in the literature. Since mathematical problem of determining a time history from given set of response spectral values is not unique, an exact solution is not possible, and all the proposed methods resort to some forms of approximate solutions [1]. A common feature of these approximate solutions is that they utilize an iterative procedure, where subsequent steps of the iteration are expected to yield better fit of the artificial time history spectrum to the predetermined target spectrum.

In this proceeding, we present the method that we use to generate response spectrum compatible artificial acceleration histories. The acceleration histories are based on the series of sinusoidal waves with specific amplitudes and the overall shape of the history is controlled by an envelope function. The compatibility of the artificial history spectrum with the target spectrum is achieved using simple amplitude correction scheme. The method, or "procedure", also incorporates some simple additional enhancements to ensure that practical artificial histories are achieved.

The procedure relays on commonly used method of numerical simulation, the one based on the fact that any periodic function can be expanded into a series of sinusoidal waves:

_{} (1)

, where *N* is the number of sinusoidal terms and *A _{i}*
is the amplitude related to circular frequency iDw. Hence, equation 1 is the Fourier series
without the constant term. The circular frequency increment is defined in our
application to be:

_{} (2)

, in which *f*^{max} is the maximum frequency
given for the target spectrum.

By fixing an array of amplitudes and then generating
different arrays of phase angles *j** _{i}*,
different motions which are similar in general appearance (i.e., in frequency
content) but different in the local details, can be generated. The procedure
uses a random number generator routine to produce strings of phase angles with
a uniform distribution in the range between 0 and 2p.

To simulate the transient character of real
earthquakes, the steady-state motions are multiplied by a deterministic
envelope function *I(t)*. The artificial motion of *z(t)* then
becomes:

_{} (3)

The definition of the envelope function is given in Fig. 1.1.

**Fig.
1.1** Envelope function for time histories.

In the Fig. 1.1, the shape of the motion is divided
in three parts. The first part, *t*Î
[0, *T _{b}*], is so called build-up time for the time history motion.
The second part,

With an envelope function shape given and a set of
phase angles selected, the remaining effort is to determine the array of
amplitudes *A _{i}* in equation 3. We followed the practice given
in Ref. [2]; an initial array of amplitudes is derived from the values of the
input target response spectrum which can be related to the Fourier amplitude
spectrum of the motion. Subsequent iterations are then performed on the basis
of a linear correction scheme in order to adjust the array of amplitudes.
Specifically:

_{}_{} (4)

_{}, where *Sa(f _{i})*

At the end of the seismic even, one would expect that the velocity and the displacement of the motion vanish. The acceleration time histories generated by equation 3, however, generally do not produce zero velocity or displacement at the end of the motion. Because of this, we apply a very simple baseline correction to the acceleration history. In this correction we introduce constant acceleration term that is scaled so that it yields zero displacement at the end of the motion.

Another problem encountered with artificial time histories, is that sometimes the peak acceleration value may be under estimated. From engineering point of view, this peak value under estimation does not really matter if the target spectrum is otherwise well matched. However, as it is generally custom to classify seismic accelerations with peak ground acceleration (PGA) values, it would be expedient to demand that the artificial acceleration history should have accurate or conservative peak value.

To tackle the peak value problem, our procedure just increases the maximum (or decreases the minimum) value of the acceleration history at a single location of the history, if the artificial acceleration history peak value seems to fall short.

Finally, to ensure that spectrum of the artificial history is conservative we have made it possible to scale the target spectra uniformly before fitting procedure is started. This is actually quite practical enhancement because, especially with low-damping spectra, the oscillations of the artificial history spectra can be notable.

To illustrate the how the iteration procedure works, we present three cases of artificial time history generation. The first two cases are ground motion spectra related and the third one can be seen as typical floor response spectrum case.

The target spectrum in this example is from Ref. [4]
and it represents the median ground response spectrum for 10^{5} years
return period for southern Finland. The target spectrum damping is 5%.

The artificial acceleration history record was defined with time increment of 1 ms. Number of sinusoidal terms in the equation 3 was set to be 1500.

Figs. 4.1(a) - 4.1(d) illustrate the spectrum generation as the iteration process advances. The Fig. 4.1(e) shows the artificial acceleration history after 20 iterations.

**Fig. 4.1(a)** Spectra comparison after first
iteration.

**Fig. 4.1(b)** Spectra comparison after 5
iterations.

**Fig. 4.1(c)** Spectra comparison after 10
iterations.

**Fig. 4.1(d).** Spectra comparison after 20
iterations.

**Fig. 4.1(e)** Artificial acceleration history after
20 iterations.

The target spectrum in this example is also from Ref. [4]. The difference from Example 1, is that the target ground spectrum is now 2% damping spectrum.

The artificial acceleration history record was defined with time increment of 1 ms. Number of sinusoidal terms in the equation 3 was set to be 1500.

Figs. 4.2(a) - 4.2(d) illustrate the spectrum generation as the iteration process advances. The Fig. 4.2(e) shows the artificial acceleration history after 20 iterations.

**Fig. 4.2(a)** Spectra comparison after first
iteration.

**Fig. 4.2(b)** Spectra comparison after 5
iterations.

**Fig. 4.2(c)** Spectra comparison after 10
iterations.

**Fig. 4.2(d)** Spectra comparison after 20
iterations.

**Fig. 4.2(e)**
Artificial acceleration history after 20 iterations.

In this example, the target spectrum is intended to represent typical floor design spectrum with broadened peak spectral values. Damping of the spectrum is 1%. The example spectrum is fictitious.

The artificial acceleration history record was defined with time increment of 1 ms. Number of sinusoidal terms in the equation 3 was set to be 2000.

Figs. 4.3(a) and 4.3(b) illustrate the spectrum generation as the iteration process advances. The Fig. 4.3(c) shows the artificial acceleration history after 30 iterations.

**Fig. 4.3(a)** Spectra comparison after first
iteration.

**Fig. 4.3(b)** Spectra comparison after 30
iterations.

**Fig. 4.3(c)**
Artificial acceleration history after 30 iterations.

The main conclusion considering the application examples are following:

1) The iteration scheme seems to converge fast. After few iterations, the gain of additional iterations is very small.

2) Artificial fittings to low-damping spectra are harder to make as they oscillate more than high-damping spectra. By using high number of sinusoidal terms in Equation 3, the quality of spectral match can be improved with low-damping target spectra.

3) The peak correction, in order to achieve given peak acceleration, clearly shows in the time histories of Example 1 and Example 2 (Figs. 4.1(e) and 4.2(e)). The peak correction is larger in low-damping case (i.e. Example 2).

4) In fictitious broadened floor spectrum Example 3, the peak acceleration value was over estimated. Otherwise, the spectral match was very high quality.

Comparing the Figs. 4.1(e) and 4.2(e), it is evident that acceleration history appearance is very different. The problem of this difference is that it is not evident which time history would be best, if the time history is used to analyze structure which has different damping than the target spectrum had. The time history of Fig. 4.1(e) will not give good match to the 2% damping spectrum and the time history of Fig. 4.2(e) will not give good match to the 5% damping spectrum.

The ideal situation would be the case where time history yields high quality match to spectra defined with multiple damping values. The multi-damping spectra methods are studied, for example, in Refs. [1], [5] and [6]. Matching the artificial history spectra to spectra defined with multiple damping values is in our endeavors in the future.

References

[1] Lilhanand K. and Tseng W.S. "Generation of synthetic time histories compatible with multiple damping design response spectra", SMiRT-9. Lausanne, Switzerland, K2/10, pp. 105-110, 1987.

[2] Xu J., Philippacopoulas A. J., Miller C. A. and Costantino C. J., "CARES (Computer Analysis for Rapid Evaluation of Structures)", Version 1.0, Theoretical Manual, Brookhaven National Laboratory, NUREG/CR-5588 BNL-NUREG-52241, Vol.1, USA, 1991.

[3] D. E. Gasparini and E. H. Vanmarcke, "SIMQKE: A Program for Artificial Motion Generation", M.I.T., 1976.

[4] Saari J., Varpasuo P. and Nikkari Y., "GROUND RESPONSE SPECTRA FOR EXTREME SEISMIC EVENTS IN SOUTHERN FILAND", The Third EU – Japan Workshop on Seismic Risk, Kyoto, Japan, March 27-30, 2000.

[5] Lilhanand K. and Tseng W.S., "Development and application of realistic earthquake time histories compatible with multiple damping design spectra" In: Proceedings of the 9th WCEE, vol. II. Tokyo-Kyoto, Japan, pp. 819-824, 1988.

[6] D. H. Choi and S. H. Lee, "Multi-damping earthquake design spectra-compatible motion histories", Elsevier, Nuclear Engineering and Design 226, pp. 221-230, 2003.