**Seismic
Hazard Assessment of Tianwan Nuclear Power Plant Site in ****China**

Pentti Varpasuo

Fortum Engineering Ltd

Absract

Ground accelerations
during strong-motion earthquakes are generally extremely irregular, resembling
random time functions; however, these irregularities show some common features.
The motion is highly oscillatory and aperiodic, it is initiated with small
amplitudes that rapidly build up until they reach an intensity that remains
almost stationary for a certain time and then decay, steadily, until the end of
the record. The 5 % bedrock field ground spectra according to HAF 0101(1) were
adopted for targets for ground motion simulation. The horizontal ground motion
spectrum was anchored to 0.2 g and the vertical ground motion spectrum was
anchored to 0.1 g. The numerical simulation method used in this study based on
the fact that any periodic function can be expanded into a series of sinusoidal
waves. The power spectral density function is generated from target response
spectra. The evolutionary character of seismic ground motion is accounted for
by using time and frequency dependent modulation of stationary motion. In the
simulation process the match of the generated motion is improved by iterative
process where the spectral density function is corrected by the response
spectrum match ratio. The match is evaluated against multiple damping target
spectra. The generated time histories shall have the following characteristics:

1. The mean of the
zero-period acceleration (ZPA) values (calculated from the individual time
histories) shall equal or exceed the design ground acceleration.

2. In the frequency
range 0.5 to 33 Hz, the average of the ratios of the mean spectrum (calculated
from the individual time history spectra) to the design spectrum, where the
ratios are calculated frequency by frequency, shall be equal to or greater than
1.

3. No one point of
the mean spectrum (from the time histories) shall be more than 10 % below the
design spectrum. When responses from the three components of motion are
calculated simultaneously on a time-history basis, the input motions in the
three orthogonal directions shall be statistically independent. Two time
histories shall be considered statistically independent if the absolute value
of the correlation coefficient does not exceed 0.3.

As a result of the
study the design acceleration time histories compliant with HAF 0101 (01)
target response spectra were generated. The compliance of generated design time
histories with HAF 0101 (01) design spectra was evaluated and shown according
to the rules enumerated in the previous section.

The limitations and
paucity of recorded acceleration time histories together with the widespread
use of time-history dynamic analysis for obtaining structural and secondary
systems’ response are the primary motivations for the development of simulation
capabilities. Individual real earthquake records are limited in the sense that
they are conditional on a single realization of a set of random parameters
(magnitude, focal depth, attenuation characteristics, frequency content,
duration, etc.), a realization that will likely never occur again and that may
not be satisfactory for design purposes.

It is the purpose of
this investigation to design a stochastic model that will generate artificial
ground acceleration time histories fitting to the pre-determined ground
response spectra.

The
derivations and terminology presented in this section follow the reference [[1]].

A
function x(t) is called deterministic if for each value t=t_{1 }it
takes a specified value x_{1 }= x(t_{1}). A function x(t) such
that for each value t=t_{1} behaves like a random variable x_{1}
= x(t_{1}) is called non-deterministic function and the ensemble of all
possible forms of x(t) define a random process with each form of x(t) being
called member of the process. This definition can be applied to records of
strong-motion earthquakes if, for example, we let x(t) be the ground
acceleration at time t after the motion begun. In this context existing records
of actual earthquakes can be considered to be members sampled from the random
process. Random process x(t) is said to be strongly stationary if its
probability law is independent of time shift t , i.e. the joint probability law of the set of random variables x(t1), x(t_{2}), ..., x(t_{n})
is the same as that of x(t_{1}+t), x(t_{2}+t), ..., x(t_{n}+t). If this joint probability law
Gaussian, the joint probability law is Gaussian, the process is said to be
Gaussian random process. In this study it will always be assumed that the mean
value of the random process x(t) is zero at all times. Thus a Gaussian process
becomes completely described by its covariance function

Equation
1 Cov_{x}(t_{1},t_{2})=E[x(t_{1})x(t_{2})]

In Equation 1
the symbol E[.] represents an average across the whole ensemble. The stationary
characteristics of the random process x(t) are reflected in that the covariance
function is not a function of both time co-ordinates, t_{1} and t_{2}
, but only on their difference t=t_{2}-t_{1}. For
stationary random process, the covariance function is called auto-correlation
function

Equation
2 R_{x} = E[x(t)x(t+t)]

Equation 2
completely describes the stochastic characteristics of the Gaussian process.
Another statistic frequently used to describe a stationary random process is
the power spectral density function

Equation
3 S_{x}(w) = 1/p _{}R_{x}(t) exp(-iwt)dt

The power spectral
density function represents the contribution of each frequency component to the
mean square value of the process.

White
noise is the mathematical idealisation of stationary random process in which
all frequencies contribute with equal intensity to the mean square value of the
process. Such process is characterised by constant power spectral density
function, S_{0}, over entire frequency range. Or, equivalently, by the
auto-correlation function

Equation
4 R(t) = pS_{0}d(t)

In Equation 4 d(t) is the Dirac delta function.

This
auto-correlation function indicates that the process has an infinite has an
infinite variance, due to contribution of entire frequency range, and is
completely non-correlated at different times. White noise can also be
interpreted as a superposition of random pulses arriving randomly in time in
which arrival times are outcomes from homogeneous Poisson process. This process
can be physically interpreted as the arrival of the train of seismic waves.

A
non-stationary counterpart of white noise is called shot noise. The physical
interpretation of the shot noise process is as a superposition of random pulses
arriving randomly in time according to non-homogeneous Poisson process. In
other words, it is a superposition of random pulses whose rate of arrivals is
random with probabilistic characteristics varying with time. These
characteristics allow the inclusion of the initial build up and decay towards
the end in the intensity of the arriving pulses. A Gaussian shot noise is fully
described by its covariance function

Equation
5 Cov(t, t+t) = f(t)d(t)

In Equation 5 f(t) is the variance intensity function of the process. In Equation 5
again the process shows an infinite variance and complete lack of correlation.
The shot noise process can be described mathematically as the product of white
noise and deterministic function of time, p(t). The variance intensity function
of the shot noise process then becomes

Equation
6 f(t) = pS_{0}p(t)^{2}

The
dynamic behaviour of a linear single degree of freedom system is governed by
the classical second order differential equation

Equation
7 _{}
+ 2zw_{0}_{}+w^{2}y = x(t)

This
equation is sometimes referred to as second order electrical filter due to its
analog in electrical circuits. The physical meaning of the parameters z and w_{0} are well known to represent the damping ratio
and the frequency of undamped vibration of the system. The response of a
mechanical system to an excitation represented by a random process is another
random process usually referred to as the output process. Formally, the output
process y(t) can be expressed in the time domain as

Equation
8 y(t) = _{}x(t)h(t-t)dt

In Equation 8
h(t) is the unit impulse response function of the filter. The response process
can also be obtained through the frequency domain in terms of the Fourier
transforms of both processes

Equation 9 Y(iw) = H(iw)X(iw)

In Equation 9
H(iw) is the complex frequency response function of
the filter. The transfer functions h(t) and H(iw) are Fourier transform pairs and either one of them is sufficient to
describe completely the steady state behaviour of the system.

If the
excitation process is Gaussian, the output process of linear system will also
be Gaussian, and fully described by its covariance function

Equation
10 Cov_{y} (t_{1}, t_{2})
= _{}Cov(t_{1 },t_{2})h(t_{1}-t_{1})h(t_{2}-t_{2})dt_{1}dt_{2}_{}

_{ }

The
system that is the target of this investigation is the free vibration of single
degree of freedom system subjected to a random motion of its support. The
excitation process x(t) represents the displacement of the support and the
output process y(t) represents the absolute displacement of the system. The
governing differential equation for this system is

Equation 11 _{} + 2zw_{0}_{} + w^{2}y = 2zw_{0}_{} + w_{0}^{2}x

The unit
impulse response transfer function for this system is

Equation
12 h(t) = w_{0}^{2}/w_{d} exp(-zw_{0}t)sin(w_{d}t + 2a)

In Equation 12 w_{d}=w_{0}(1-z^{2})^{1/2} and a = arcsin(z).

The complex
frequency response transfer function is

Equation 13 H(iw) = (1 + 2ziw/w_{0})/(1-w^{2}/w_{0}^{2} + 2izw/w_{0})

The derivations and
terminology presented in this section follow the reference [[2]].

The random
fluctuations observed in records of strong-motion earthquakes usually follow
general patterns that can be used in the design of a stochastic model to
simulate their effect. They begin with small amplitudes, which increase in time
until the stronger shocks occur. When the main shocks are over, the amplitudes
decay steadily until the motion ends. It is also observed that real earthquakes
do not show components in every frequency. Rather they show predominant
frequencies within a relatively narrow hand. Earthquakes recorded on firm soil
at moderate epicentral distances show predominant frequencies in the range of
2-5 Hz. These characteristics have led investigators to design various types of
stochastic* *processes for the
generation of pseudo-earthquakes.

The power spectral
density function of filtered white noise can be written in form

Equation
14 S_{x}(w) = S_{0}|H(iw)|^{2}^{}

In Equation 14 H(iw)
is the complex frequency response function of the filter and S_{0 }is
the power spectral density of the white noise excitation. For a second order
linear filter as described in section 2.3, the power spectral of the output
process becomes

Equation 15 S_{y}(w) = S_{0} (1 + 4z^{2}w^{2}/w^{2})/((1-w^{2}/w_{0}^{2})^{2} + 4z^{2}w^{2}/w_{0}^{2})

The shot
noise model appropriate for ground motion simulation is obtained by filtering a
shot noise through a second order linear filter. Both the filter and the shot
noise is selected so that relevant features of strong-motion records at
moderate epicentral distances.

A
Gaussian non-stationary shot noise is used to represent the acceleration at
bedrock during an earthquake. This process simulates the effect of random
pulses arriving as seismic waves to the bottom of a superficial soil layer. The
non-stationary character shows the variability in the random arrival times of these
seismic waves. This stochastic process is completely characterized by f(t), the variance intensity function of the shot noise.

The
second order linear filter, used in this investigation, represents a damped
single degree of freedom system formed by a mass supported on a spring and a
dashpot in parallel. The acceleration at bedrock is used as the acceleration of
the support and the absolute acceleration of the mass simulates the
acceleration of the ground surface. Roughly, this model simulates the
fundamental shear mode response of the ground layer. The behavior of the
mechanical model is completely defined by the parameters z and w_{0}.

The
acceleration at bedrock is simulated by a Gaussian shot noise obtained as the
product of a stationary white noise, w(t), of intensity S_{0 }and a
deterministic function of time, p(t); thus,

Equation
16 a_{b}(t) = p(t)w(t)

The shaping
function, p(t), must be zero for negative arguments, nonnegative e1sewhere,
and, in general, a smoothly varying function of time. It is related to the
variance intensity function by Equation 6. The
filtering effect of a soil layer is simulated by a damped single degree of
freedom linear system with damping ratio
z and an undamped circular frequency w_{0}.The ground
acceleration a_{g}(t) is obtained as the absolute acceleration of the
mass from the set of equations

Equation
17 _{}
+ 2zw_{0}_{}
+ w_{0}z = - a_{b}(t)

Equation
18 a_{g}(t)
= -(2zw_{0}_{}
+ w_{0}^{2}z)

If initial
conditions of zero ground velocity and zero ground displacements are assumed,
the ground acceleration process can be formally expressed as

Equation
19 a_{g}(t) = _{} a_{b}(t) h(t—t) dt

In Equation 19 the unit
impulse response function of the filter, h(t), is given in Equation 12. Through
the frequency domain, the ground acceleration process can be obtained in terms of its Fourier transform, as

Equation
20 A_{g}(iw) = H(iw)A_{b}(iw)

In Equation 20 complex
frequency response function, H(iw) is given in Equation 13. The
stochastic characteristics of the ground acceleration process are represented
by the covariance function obtained from Equation 10 and Equation 5, i.e.

Equation
21 Cov(t_{1},t_{2})
= _{}f(t)h(t_{1} - t)h(t_{2} - t)dt

For the particular
case of t_{1} = t_{2} , this covariance function becomes the
variance function of the process, namely

Equation
22 s^{2}(t) = _{}f(t)h(t-t)^{2}dt

It can be
conclude that the ground acceleration process is Gaussian and is completely
defined by filter parameters, z and w_{0 }, and by
f(t), the variance intensity function of the
shot noise.

The
filter parameters and the variance intensity function of the shot noise are
estimated using available earthquake records that are assumed to be sample wave
forms generated by the model. In order to obtain significantly consistent
results, only records of earthquakes occurring in similar conditions are used.
Thus the model will actually simulate earthquake records corresponding to these
conditions of soil, epicentral distance, range of magnitude, etc. The records
used in the example are the two horizontal components of ground acceleration
recorded during the earthquakes:

1. El Centro, California,
December 12, 1934, N and W;

2. El Centro, California,
May 18, 1940, N and W;

3. Olympia, Washington,
April 13, 1949, N80E N10W;

4. Taft, California, June
21 1952, N69W and S21W.

These
records were normalized to a unit spectral intensity as defined by Housner in
reference [[3]] and
are assumed to be waveforms sampled from the ground acceleration process.

The Fourier
amplitude transforms of the excitation and output processes of the model are
related by Equation 20 as

Equation
23 |A_{g}(iw)|^{2 } = |H(iw)|^{2}|A_{b}(iw)|^{2}^{}

Averaging Equation 23 across
the whole ensemble and using the definition of the bedrock acceleration process
Equation 16 , the
ensemble average becomes from Equation 5

Equation
24 E[|A_{b}(iw)|^{2}] = _{}f(t)dt = const.

Thus, it may be seen
that the ensemble average across the ground acceleration process in Equation 23 has the
same shape as the frequency response amplitude, namely

Equation
25 E[|A_{g}(iw)|^{2}] = |H(iw)|^{2} *const

This relation is
used in the estimation of the filter parameters by computing E[|A_{g}(iw)|^{2}]
as an average of sample wave forms and estimating the values of z
and w_{0 }that give the best fit with

Equation
26 |H(iw)|^{2} = (1 + 4z^{2}w^{2 }/w_{0}^{2})/[(1-w^{2}/w_{0}^{2})^{2 }+4z^{2}w^{2}/w_{0}^{2}]

Letting F_{i}(iw)
be the Fourier transform of the sample wave form f_{i}(t), the ensemble
average is estimated as

Equation
27 E[|A_{g}(iw)|^{2}] = 1/n _{}|F_{i}(iw)|^{2}^{}

The best fit was
found with parameters z = 0.6 and w_{0} = 5p

The variance
intensity function is estimated from Equation 22 . The
Fourier transform of Equation 22 gives

Equation
28 S^{2}(iw) = F(iw) K(iw)

In Equation 28 K(iw)
is the Fourier transform of function
h(t)^{2}. Substituting the computed expressions for S^{2}(iw)
and K(iw) into Equation 28 we
obtain an estimate for F(iw), the Fourier transform of the variance
intensity function, i.e.

Equation
29 F(iw) = S^{2}(iw)/K(iw)

An inverse Fourier
transform provides an estimate of f(t). The smooth variance intensity fitted to
the sample wave form can written as

Equation
30 f(t) = 5.11*10^{-5} sec 0<t<11.5 sec

f(t) = 5.11*10^{-5 }sec *exp(-0.155
sec^{-1}(t-11.5 sec)) t>11.5
sec

First, samples of
white noise are generated, then these samples are shaped using the shaping
function and passed through the filter to obtain the wave forms representing
ground acceleration.

A sequence of
independent random numbers u_{j }with uniform distribution in the
interval (0,1) is generated. A new sequence of independent random numbers w_{j
}with Gaussian distribution having zero mean and unit variance is obtained
with transformation

Equation
31 w_{j}
= (-2 ln u_{j})^{1/2}cos
(2pu_{j+1}) j=odd

Equation
32 w_{j+1}
= (-2 ln u_{j })^{1/2}sin (2pu_{j+1}) j=odd

This sequence of
white numbers is now spaced at intervals Dt with the origin time
randomly sampled from uniform distribution in the interval (0,Dt). Repeating this procedure a sufficient number of times, an ensemble of
random waveforms w(t) is obtained. If the ordinates w_{j }of each of
wave forms are multiplied by (pS_{0}/Dt)^{1/2}
, the autocorrelation of the process becomes

Equation
33 R(t) = 0 if t<2Dt

R(t) = pS_{0}/Dt(4/3 +
2(t/Dt) + 1/6(t/Dt)^{3}) if -2Dt<t<-Dt

R(t) = pS_{0}/Dt(2/3 -
(t/Dt)^{2} -1/2(t/Dt)^{3}) if -Dt<t<0

R/t) = pS_{0}/Dt(2/3 -
(t/Dt)^{2} + 1/2(t/Dt)^{3}) if 0<t<Dt

R(t) = pS_{0}/Dt(4/3 -
2(t/Dt) + 1/6(t/Dt)^{3}) if Dt<t< 2Dt

R(t) = 0 if
t< 2Dt

Equation 33
in the limit, as Dt approaches zero, approaches the
form

Equation
34 R(t)
= pS_{0}d(t)

This
limiting case corresponds to a white noise with constant power spectral density
S_{0.}

The
non-stationary shot noise chosen to represent the bedrock acceleration process
is obtained multiplying a white noise of intensity S_{0} times a
shaping function p(t). The shaping function p(t) is obtained in terms of
variance intensity function as follows

Equation
35 p(t)
= [f(t)/p/S_{0}]^{1/2}^{}

For purposes of
numerical calculations, the shaping function can be lumped with the scaling
factor. Thus bedrock acceleration wave forms become

Equation
36 a_{b}(t)
= (f(t)/Dt)^{1/2}w(t)

The waveforms of the
bedrock acceleration ensemble were filtered trough second order linear filter
to obtain the ground acceleration a_{g}(t) from the differential
equation

Equation
37 _{}
+ 2zw_{0}_{}
+ w_{0}^{2} z = -a_{b}(t)

a_{g}(t) = -2zw_{0}_{} - w_{0}^{2}z

The solution of this
equation is obtained step-by-step procedure with piecewise linear acceleration
assumption.

The derivations and methodology described in this section follow the reference [[4]].

The 5%
bedrock field ground spectra according to HAF 0101(1) were adopted for targets
for ground motion simulation. The horizontal ground motion spectrum was
anchored to 0.2g and the vertical ground motion spectrum was anchored to 0.1g.

In the
following Table 1 the spectral ordinates as functions of frequency are given in
tabular form:

Horizontal
spectrum |
||||||

Frequency |
0.25 HZ |
3.3 HZ |
14.3 HZ |
25 HZ |
33.3 HZ |
50 HZ |

Horizontal |
0.062g |
0.610g |
0.538g |
0.346g |
0.2g |
0.2g |

Vertical spectrum |
||||||

Frequency |
0.25 HZ |
4 Hz |
14.3 Hz |
25 Hz |
33.3 Hz |
50 HZ |

Vertical |
0.084g |
0.262g |
0.294g |
0.182g |
0.1g |
0.1g |

Table 1 Tabulated
values of vertical and horizontal design spectra

The
limitations and paucity of recorded acceleration time-histories together with
the widespread use of time-history dynamic analysis for obtaining structural
and secondary systems' response are the primary motivations for the development
of simulation capabilities. Individual real earthquake records are limited in
the sense that they are conditional on a single realisation of a set of random
parameters (magnitude, focal depth, attenuation characteristics, frequency
content, duration, etc.), a realisation that will likely never occur again and
that may not be satisfactory for design purposes.

The
intent herein is to focus on one 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:

Equation
38 x(t)
= _{}A_{i}
sin(w_{i} t + f_{i})

A_{i} is the
amplitude and f_{i} is the phase angle of the ith contributing sinusoid. By fixing an array
of amplitudes and then generating different arrays of phase angles, different
motions which are similar in general appearance (i.e., in frequency content)
but different in the local details, can be generated. The computer uses a
"random number generator" subroutine to produce strings of phase
angles with a uniform distribution in the range between 0 and 2p. The
total power of the steady state motion, x(t) is _{}(A^{2}_{i}/2). Assume now that the frequencies w_{i }in Equation 38
are chosen to lie at equal intervals Dw. Figure 3
shows a function G(w) whose value at w_{i} is equal to A_{i}^{2} /2Dw so that G(w_{i})Dw = A_{i}^{2}/2.
Allowing the number of sinusoids in the motion to become very large, the total
power will become equal to the area under the continuous curve G(w), which is in effect the spectral density function. When G(w) is narrowly
centred around a single frequency, then Equation 38 will generate nearly sinusoidal functions. On
the other hand, if the spectral density function is nearly constant over a wide
band of frequencies, components with widely different frequencies will compete
to contribute equally to the motion intensity, and the resulting motions will
resemble portions of earthquake records. Of course, the total power and the
relative frequency content of the motions produced by using Equation 1 do not
vary with time. To simulate in part the transient character of real
earthquakes, the stationary motions, x(t), are usually multiplied by a
deterministic intensity function such as the exponential compound function.

The most
widely used equation form for power spectral density function is so called
filtered white-noise spectral density function expressed by Equation 2 is as
follows:

Equation
39 _{}

_{}

In Equation
39 the filtered white noise power spectral density function is depicted with
the following arbitrary parameters: G_{0}=100, w_{g}=50, z_{g}=0.5. In actual simulation task the parameters
are so selected that the power spectral density function corresponds as closely
as possible the given target response spectrum. Theoretical spectral density
shapes such as Equation 39
are obtained by examining, smoothing and averaging of the squared Fourier
amplitudes |f(w)|^{2} of actual strong
motion records. This stems from the basic fact that, for stationary random
processes, the expected value of |f(w)|^{2 }and
the spectral density functions G(w) are proportional.

In
summary, the problem in simulation with the model represented by Equation 38
is, then, one of selecting a proper shape and intensity of either the power spectral
density function or the Fourier amplitude spectrum and estimating the duration
of the motion. Development of design criteria has evolved along different
paths, however. In the nuclear industry, for example, a set of smooth response
spectra has been adopted for use in seismic design. In this practice the power
spectral density function is generated from smooth target response spectra. In
the simulation generation process the match of the generated motion is improved
by iterative process where the spectral density function is corrected by the
response spectrum match ratio as follows:

Equation 40 G(w)_{i+1} = G(w)_{i} (S^{t}(w)/S^{i} (w))

In Equation 40
S^{t}(w) is the target response spectral
value.

The
generated time histories shall have the following characteristics:

1. The mean of the zero-period acceleration (ZPA) values (calculated from the
individual time histories) shall equal or exceed the design ground
acceleration.

2. In the frequency range 0.5 to 33 Hz, the average of the ratios of the mean
spectrum (calculated from the individual time history spectra) to the design
spectrum, where the ratios are calculated frequency by frequency, shall be
equal to or greater than 1.

3. No one point of the mean spectrum (from the time histories) shall be more
than 10% below the design spectrum. When responses from the three components of
motion are calculated simultaneously on a time-history basis, the input motions
in the three orthogonal directions shall be statistically independent. Two time
histories shall be considered statistically independent if the absolute value
of the correlation coefficient does not exceed 0.3.

The
generated time histories and corresponding response spectra together with
target spectra are depicted in following figures:

Figure 1 Horizontal x-axis ground acceleration time history

Figure 2 Horizontal x-axis ground response spectrum fit

Similar
figures can be constructed for y-axis and vertical acceleration time histories
and target ground response spectra.

According
to reference [[5]] the input motion time histories
should meet the following requirements:

1. One or more recorded, modified recorded or synthetic earthquake ground motion time histories may be used to calculate seismic response of safety related structures.

2. Time histories shall be so selected or developed that they reasonably represent the duration of strong shaking conditions expected for the site. For analysis shorter time segments may be used if they satisfy the conditions given below:

3. The developed time histories shall have following characteristics:

· the mean of the zero-period acceleration (ZPA) calculated from the individual time histories shall equal or exceed the design ground acceleration

· in the frequency range 0.5 t0 33 Hz the average of the ratios of the mean spectrum ( calculated from the individual time history spectra) to the design spectrum, where the ratios are calculated frequency by frequency shall be equal, shall be equal to or greater than 1.

· no one point of the mean spectrum (from the time histories shall be more than 10% below the design spectrum

Spectral values from the time history shall be calculated at sufficient frequency intervals to produce accurate response spectra. The following table provides suggested frequencies at which spectral ordinates may be calculated:

Frequency range (Hz) |
Increment(Hz) |

0.5-3.0 |
0.10 |

3.0-3.6 |
0.15 |

3.6-5.0 |
0.20 |

5.0-8.0 |
0.25 |

8.0-15.0 |
0.50 |

15.0-18.0 |
1.0 |

18.0-22.0 |
2.0 |

22.0-34.0 |
3.0 |

Table 2 Suggested frequencies for calculating of response according to reference [5].

Another
acceptable method to choose the frequencies is such that each frequency is
within 10% of the previous frequency. When responses from the three components
of the motion are calculated simultaneously on a time history basis the input
motion in the orthogonal directions shall be statistically independent, and the
time histories shall be different. Shifting the starting time of single time
history does not constitute the establishment of different time history. Two
time-histories shall be considered statistically independent if the absolute
value of the correlation coefficient does not exceed 0.3.

Correlation
coefficients between synthetic excitation components were developed using EXCEL
[[6] ] data
analysis software. The resulting matrix of correlation coefficients is given in
the following table:

Table 3 Table of correlation coefficients between synthetic excitation components

As can be
seen from the Table 3 that the correlation coefficients between vertical and
horizontal components are less than 0.05 and the correlation coefficient
between two horizontal components is 0.18. All these coefficients are less than
0.3 and the requirement of statistical independence of components is fulfilled.

Summary
statistics of the time series representing the earthquake components are given
in the following table:

Table 4 Summary statistics of synthetic earthquake excitation components

From the Table 4 it is seen that the ZPA values of the
components are in g units are 0.2001447 for both horizontal components and
0.100 for vertical component and so the first requirement of reference [5] is fulfilled.

The fit
to reference [[7]] target spectrum corresponding to 5%
damping is given the following table:

Table 5 Fit to HAF 0101 (01) 5% damping target spectra

As it can
be seen from Table 5
all ASCE 4-86 spectral fit criteria all fulfilled for synthetic time histories
developed.

The match
of Tianwan Nuclear Power Plant design acceleration time histories with HAF 0101
(01) Design spectra is studied. The compliance of design time histories with
HAF 0101 (01) design spectra according to ASCE 4- 86 [5]
rules are shown.

[[1]] Y.K. Lin, "Probabilistic theory of structural dynamics", Robert E. Krieger Publishing Co., Florida, 1967.

[[2]] P. Ruiz and J. Penzien, " PSEQN - Artificial generation of earthquake accelerograms", Earthquake Engineering Research Center, Report No EERC 69-3, University of California, Berkeley, California, March 1969.

[[3]] Housner G.W., "Behavior of Structures During Earthquakes, Journal of Engineering Mechanics Division, ASCE, 85, No EM4, pp 109-129, October 1969.

[[4]] D. A. Gasparini, E. H. Vanmarcke, "Simulated earthquakes compatible with prescribed response spectra", Massachusetts Institute of Technology, Publication No. R76-4, Cambridge, Massachusetts, January 1976.