Research on Time history Input Methodology of Seismic Analysis

Research on Time-history Input Methodology of Seismic Analysis

Jiang Naibin, Mao Qing and Zhang Yixiong

State Key Laboratory of Reactor System Design Technology, Nuclear Power Institute of China, Chengdu, China

ABSTRACT

Several methods exist to input seismic excitation when making the seismic time-history analysis for design of nuclear power plant: 1) to input the displacement time-history at the base, which is called displacement method; 2) to input the inertia loading calculated from the time-history of support motion acceleration, which is called acceleration method; 3) large mass method. This paper deduced the theoretical basis of the first two methods, and presented a new method based on acceleration method, called modified acceleration method. Through a numerical example, the seismic responses with three methods have been compared respectively, while the restriction conditions for using the three methods were discussed. Seismic analysis of reactor coolant system of a three-loop nuclear plant was also carried out with above-mentioned three methods, and three different response results were given.

INTRODUCTION

As a major natural hazard, earthquakes may cause extremely damaging for industrial or power-generating facilities. To avoid the potential harm of earthquakes, strict safety rules have been established in the design of nuclear power plant. These seismic safety rules can be classified into two levels: the Operational Basis Earthquake (OBE) for maintaining unit operability for a seismically probable level, on site-specific historical and geological bases, and the Safe Shutdown Earthquake (SSE) for returning of the reactor to a permanent shutdown condition. During the earthquake evaluation, safety-related structures, systems and components classified as seismic category I should withstand the loads imposed by these two hypothetical earthquakes.

Time-history and response spectrum methods are the two basic methods that are commonly used for the seismic dynamic analysis. The time-history method is relatively more time consuming, lengthy and costly. The response spectrum method, on the other hand, is relatively more rapid, concise, and economical. However, time-history method must be employed when geometrical and/or material non-linearities are present in the structural systems. Nowadays, it’s more convenient for using time-history method than before for advancing of computer’s hardware and software.

Several methods exist to input seismic excitation when making the seismic time-history analysis for design of nuclear power plant: 1) to input the displacement time-history at the base, which is called displacement method; 2) to input the inertia loading calculated from the time-history of support motion acceleration, which is called acceleration method; 3) large mass method. This paper deduced the theoretical basis of the first two methods, and presented a new method based on acceleration method, called modified acceleration method. Through a numerical example, the seismic responses with three methods have been compared respectively, while the restriction conditions for using the three methods were discussed. Seismic analysis of reactor coolant system of a three-loop nuclear plant was also carried out with above-mentioned three methods, and three different response results were presented.

METHODS OF SEISMIC INPUT IN TIME-HISTORY ANALYSIS

Displacement Method

For a three-dimensional structural system with seismic excitation at supports, coupled equations of motion can be written in the partitioned matrix form as:

⎭

⎬

⎫

⎩

⎨

⎧

=

⎭

⎬

⎫

⎩

⎨

⎧

⎥

⎦

⎤

⎢

⎣

⎡

+

⎭

⎬

⎫

⎩

⎨

⎧

⎥

⎦

⎤

⎢

⎣

⎡

+

⎭

⎬

⎫

⎩

⎨

⎧

⎥

⎦

⎤

⎢

⎣

⎡

B BB T AB AB AA BB T AB AB AA BB T AB AB AA

F

0

Z

X

K

K

K

K

Z

X

C

C

C

C

Z

X

M

M

M

M

&

&

&&

&&

(1)

where a dot over a quantity denotes differentiation with respect to time and the superscript “T” denotes transpose of a matrix. X is a N1×1 vector of unknown absolute translational displacements at the N1 number of nonsupport degrees of

freedom. Z is a N2×1 vector of absolute translational displacements at the N2 number of support degrees of freedom. M,

C and Kdenote mass matrix, damping matrix and stiffness matrix, respectively. The submatix with subscript “AA” is associated with X,

X

&

or

X

&&

, and “BB” with Z,

Z

&

or

Z

&&

. The subscript “AB” denotes the coupling term.

When we use displacement method, the displacement time-history of support motion is known as the boundary condition of Eq. (1). If the initial condition is given, the absolute translational displacements X can be obtained by solving Eq. (1). As we care more about dynamic displacement parts than the whole absolute displacements in seismic analysis, so the output data of displacement response has to be reprocessed. In addition, seismic excitation is usually recorded in the form of time-history of ground motion acceleration, therefore, displacement time-history has to be obtained from acceleration time-history through two times of time integral. This process will introduce a bit of numerical error.

Acceleration Method

The first set of equations in Eq. (1) may be rewritten as follows:

Z

K

Z

C

Z

M

X

K

X

C

X

M

_AA

&&

+

_AA

&

+

_AA

=

−

_AB

&&

−

_AB

&

−

_AB (2)

Eq. (2) represents a set of N1 equations for the unknown absolute displacements X. The right hand side of Eq. (2)

represents a N1×1 vector of known forcing functions since the motion (i.e., Z, and ) is specified at the support

degrees of freedom.

Z

&

Z

&&

The displacement vector X and Z can be decomposed into static and dynamic parts using the following definition:

s d

X

X

X

=

+

(3a)

s d

Z

Z

Z

=

+

(3b)

where Xd is a N1×1 vector of dynamic displacements contributing to X; Xs is a N1×1 vector of static displacements

contributing to X; Zd is a N2×1 vector of dynamic displacements contributing to Z (Note that this vector is identically

equal to a null vector since Z is specified apriori); Zs is a N2×1 vector of static displacements contributing to Z ( Note

that this vector is identically equal to Z since Zd = 0 ).

The static displacements, Xs, can be calculated from Eq. (2) by setting mass and damping matrices equal to zero. Thus,

one obtains

Z

K

K

X

1 _AB

AA

−

−

=

s (4)

The above equation defines the time-varying equilibrium position Xs (t) of the system under the imposed displacements

Z (t).

Substituting Eq. (3) into Eq. (2) and utilizing Eq. (4), the following equation is obtained:

Z

K

K

C

C

Z

K

K

M

M

X

K

X

C

X

M

&&

&

[

]

&&

[

AB

]

&

1 AA AA AB AB 1 AA AA AB d AA d AA d AA − −

₊

₋

−

=

+

+

(5)

where MAB is identically equal to a null matrix for a lumped mass formulation; can be replaced by the

matrix R. Thus, Eq. (5) can be simplified as:

AB 1 AA

K

K

−

−

Z

R

C

C

Z

R

M

X

K

X

C

X

M

AA

&&

d

+

AA

&

d

+

AA d

=

−

AA

&&

+

[

AB

−

AA

]

&

(6)

in the most time. Therefore, it can be neglected [1], and Eq. (6) can reduces to the basis equation of acceleration method:

Z

R

M

X

K

X

C

X

M

_AA

&&

_d

+

_AA

&

_d

+

_{AA d}

=

−

_AA

&&

(7)

For acceleration method, Eq. (7) is used for time-history analysis of structural systems subject to uniform excitation at supports. Comparing with displacement method, this method has no need to do any reprocessing with the input and output data.

Modified Acceleration Method

If the second term on the right hand side of Eq. (6) was taken into account, a new method for seismic input in time-history analysis can be obtained. If one makes the assumption of Rayleigh damping, i.e.:

AA AA

AA

M

K

C

=

α

+

β

(8a)

AB AB

AB

M

K

C

=

α

+

β

(8b)

Thus, Eq. (6) may be rewritten as follows:

E

Z

R

M

X

K

X

C

X

M

_AA

&&

_d

+

_AA

&

_d

+

_{AA d}

=

−

_AA

&&

(9a)

Z

Z

Z

&&

_E

=

&&

+

α

&

(9b)

For modified acceleration method, the velocity, , can be obtained from by time integral, then the equivalent acceleration can be calculated from Eq. (9b). Other processes are same as the acceleration method, except replacing with . Obviously, the numerical error introduced by preprocess excitation data in this method is smaller than the one introduced in displacement method. Like acceleration method, this method can be used only for time-history analysis of structural systems subject to uniform excitation at supports.

Z

&

Z

&&

E

Z

&&

Z

&&

Z

&&

_E

NUMERICAL EXAMPLE

For comparing the three methods mentioned above, a sample numerical example is presented. Three mass-spring systems are connected to the ground (see Fig. 1). Fig.2 shows the time-history of ground acceleration, and model parameters are specified in Tab. 1.

0 5 10 15 20 25 30 35

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8

ac

celeratio

n

/m

s

-2

time /s

Table 1. Model parameters

Mass / kg Stiffness of springs / kN.m-1

Rayleigh damping constants

M1 M2 M3 K1 K2 K3 α β

1000 1000 1000 200 7885 265761 2.326 0.000295

Table 2. The maximum load in springs for three methods of excitation input

Displacement method Acceleration method Modified acceleration method

Force / N 1628 1507 1626

Mass-spring

system 1 Time / s 1.97 1.96 1.97

Force / N 1709 1455 1634

Mass-spring

system 2 Time / s 2.70 2.69 2.70

Force / N 856.4 657.2 843.0

Mass-spring

system 3 Time / s 4.02 2.79 4.02

For three methods of excitation input, three sets of response results can be obtained by using ANSYS 8.0 [2] (see Tab. 2). The differences of results between the displacement method and the modified acceleration method are relatively small, in that they all have considered the second term on the right hand side of Eq. (6). The responses obtained in acceleration method and in modified acceleration method are different because of the different excitation input.

SEISMIC ANALYSIS ON REACTOR COOLANT SYSTEM OF A THREE-LOOP NUCLEAR PLANT

Nonlinear analytical modeling

With above-mentioned three methods, a seismic analysis on reactor coolant system of a PWR plant was carried out.

The system consists of the Reactor Pressure Vessel (RPV) and three loops, each comprising a Steam Generator (SG), a Reactor

Coolant Pump (RCP), and the reactor coolant pipes. The pressurizer is connected to one of the loops through surge line. The

nonlinearities in the reactor coolant system mainly appear at the support position. The RPV is supported by six support pads that fit

into recesses in a circular support ring mounted on a ledge inside the reactor pit. This configuration allows the RPV to move freely

upwards, but will lock the downwards movement. The support legs and snubbers of the equipment and surge line have bilinear

stiffness characteristics. The SG and pressurizer are supported laterally by the stops which maintain 1 to 4.4mm gaps between the

equipment shells, also with the inelastic stress-strain relationship at the SG lower lateral stops.

The primary equipment is simulated by three-dimensional beam element. The lumped mass points modeling internals attach on the beam element. Three kinds of pipe elements are used according to the structure form of the primary piping and surge line: elastic straight pipe, elastic pipe tee and elastic curved pipe (elbow). The reactor building internal structure is modeled by a stick model with three-dimensional beam and lumped mass, representing the civil walls and certain floors. Many kinds of support are included in the reactor coolant system such as support skirt, support leg, support ring, stop, snubber, and pipe whip restraint. Generally, the linear supports are modeled by linear spring. The nonlinearities in the supports including mutilinear stiffness, gap, and inelastic are stimulated by nonlinear spring. Fig.3

shows the analytical model of the reactor coolant system.

damping ratios,

ξ

_i. If

ω

_i is the natural circular frequency of mode i,

α

and

β

satisfy the relation:

2

2

i i i

βω

ω

α

ξ

=

+

(10)

Fig.3 Analytical model of the reactor coolant system

Seismic input

An artificial time-history is generated from the design response spectra (US NRC RG1.60) at free field for carrying out a time-history analysis. The time history responses can be obtained from each of the three components of the earthquake motion, two horizontal and one vertical directions, and combined at each time step algebraically. The maximum response is then derived from the combined time solution. Using this method, the three components of the artificial time-history should be statistically independent.

Results

The seismic analysis under SSE has been fulfilled using finite element code ANSYS8.0. The maximum responses at the

typical location are presented in Table 3. For the complexity induced by nonlinearities in the system, any evident conclusion can not obtained by comparing the results for three methods of excitation input. But, as a whole, the response results for modified acceleration method are closer with those for displacement method.

DISCUSSION

If one makes the assumption of Rayleigh damping, the second term on the right hand side of Eq. (6) reduces to which would vanish if

Z

R

M

A

&

α

acceleration method may have some limitation, so displacement method and modified acceleration method should be adopted, and if support excitation is recorded in the form of acceleration time-history, modified acceleration method would be better for decreasing numerical error introduced during process the input data.

Table 3. The response results at typical location for three methods of excitation input

Displacement

method Acceleration method

Modified acceleration method

Minimum Maximum Minimum Maximum Minimum Maximum

Horizontal displacement

at steam nozzle / mm -15.06 16.79 -12.81 16.35 -15.45 16.81

Torsional moment

/104_N.m -15.45 6.31 -15.21 5.78 -15.51 6.25

Elbow at the

SG inlet Bending moment

/104N.m -14.07 7.01 -14.38 7.06 -13.37 6.98

Torsional moment

/104N.m -11.27 7.16 -9.76 6.48 -12.57 8.37

Elbow at the

SG outlet Bending moment

/104N.m -46.46 26.79 -49.20 28.23 -46.50 33.25

Axial force

/104N -65.78 67.02 -67.75 72.01 -67.97 68.81

Lateral force

/104_N -11.47 19.64 -10.57 22.29 -9.95 18.91

Torsional moment

/104N.m -15.18 6.14 -14.42 5.84 -15.17 6.01

RPV Outlet

nozzle

Bending moment

/104N.m -34.15 38.62 -32.75 43.50 -31.50 43.53

Horizontal loads

/104N -73.35 90.97 -74.22 104.90 -75.49 93.67

Support at

RPV inlet Vertical loads

/104_N -44.22 47.20 -41.57 48.92 -52.43 44.42

Horizontal loads

/104_N -68.26 75.49 -69.47 73.49 -70.93 78.62

Support at

RPV outlet Vertical loads

/104N -41.64 42.53 -36.68 50.31 -45.67 47.96

REFERENCES:

[1] Clough, R. W. Recent Advances in Matrix Methods of Structural Analysis and Design, University of Alabama Press, 1971.

[2] ANSYS, Inc, ANSYS User’s manual for Revision 8.0, 2003.

[3] US NRC RG1.61, “Damping Values for Seismic Analysis for Nuclear Power plants”, 1973.

[4] Hong Jing-fen. “Overview of Seismic Analysis for Nuclear Power Plant,” Nuclear Power Engineering, 17 (3), 1996, pp. 193-198. (in Chinese)