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Transactions, SMiRT 19, Toronto, August 2007 Paper # K13/1

Research on - Input of Seismic Analysis

Jiang Naibin, Mao Qing and Zhang Yixiong

State Key Laboratory of Reactor System Design Technology, Nuclear 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 : 1) to input the time-history at the base, which is called displacement method; 2) to input the loading calculated from the time-history of support , which is called acceleration method; 3) large 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 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 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 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 can be written in the partitioned matrix form as: Transactions, SMiRT 19, Toronto, August 2007 Paper # K13/1

⎡ AA MM AB ⎤⎧X&& ⎫ ⎡ AA CC AB ⎤⎧X& ⎫ ⎡ AA KK AB ⎤⎧X⎫ ⎧ 0 ⎫ ⎢ T ⎥⎨ ⎬ + ⎢ T ⎥⎨ ⎬ + ⎢ T ⎥⎨ ⎬ = ⎨ ⎬ (1) ⎣ AB MM BB ⎦⎩Z&& ⎭ ⎣ AB CC BB ⎦⎩Z& ⎭ ⎣ AB KK BB ⎦⎩Z⎭ ⎩FB ⎭ where a dot over a 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 K denote mass matrix, 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 . 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 . This process will introduce a bit of numerical error.

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

AA AA &&& AA AB AB &&& −−−=++ ABZKZCZMXKXCXM (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, Z& and Z&& ) is specified at the support degrees of freedom. The displacement vector X and Z can be decomposed into static and dynamic parts using the following definition:

= + XXX sd (3a)

= + ZZZ sd (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

−1 s −= AA ABZKKX (4)

The above equation defines the time-varying equilibrium 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:

−1 −1 dAA &&& dAA dAA [ AB −=++ AAAA AB && [] AB −+ AAAA AB ]ZKKCCZKKMMXKXCXM & (5)

−1 where MAB is identically equal to a null matrix for a lumped mass formulation; − AAKK AB can be replaced by the matrix R. Thus, Eq. (5) can be simplified as:

dAA &&& dAA dAA AA && [ AB −+−=++ AA ]ZRCCZRMXKXCXM & (6)

The term on the right hand side of the above equation is small in comparison with the first term on the right hand Transactions, SMiRT 19, Toronto, August 2007 Paper # K13/1 in the most time. Therefore, it can be neglected [1], and Eq. (6) can reduces to the basis equation of acceleration method:

dAA &&& dAA dAA −=++ AA ZRMXKXCXM && (7)

For acceleration method, Eq. (7) is used for time-history analysis of structural systems 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 = α AA + βKMC AA (8a)

AB = α AB + βKMC AB (8b)

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

dAA &&& dAA dAA −=++ AA ZRMXKXCXM && E (9a)

E += αZZZ &&&&& (9b)

For modified acceleration method, the , Z& , can be obtained from Z&& by time integral, then the equivalent acceleration Z&& E can be calculated from Eq. (9b). Other processes are same as the acceleration method, except replacing Z&& with Z&& E . 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.

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.

0.8

0.6

0.4 -2 0.2 /ms

0.0

-0.2 acceleration -0.4

-0.6

-0.8 0 5 10 15 20 25 30 35 time /s

Fig. 1 Analytical model Fig. 2 Time-history of ground acceleration

Transactions, SMiRT 19, Toronto, August 2007 Paper # K13/1

Table 1. Model parameters

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

Modified acceleration Displacement method Acceleration method method Mass-spring / N 1628 1507 1626 system 1 Time / s 1.97 1.96 1.97 Mass-spring Force / N 1709 1455 1634 system 2 Time / s 2.70 2.69 2.70 Mass-spring Force / N 856.4 657.2 843.0 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. The specific critical damping ratios are 2% (OBE) and 4% (SSE) for the reactor coolant system, with 5% (OBE) and 7% (SSE) for the reactor building internal structure[3]. Rayleigh damping constants can be calculated from modal Transactions, SMiRT 19, Toronto, August 2007 Paper # K13/1

damping ratios, ξi . If ωi is the natural circular of mode i, α and β satisfy the relation:

α βωi ξi += (10) ωi 22

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 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 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

− α A ZRM & which would vanish if α = 0 irrespective of the of β . For most condition in nuclear engineering, lightly damped systems are considered, so acceleration method can be adopted [4]. But in some special conditions, such as analysis of structural systems which are immersed in reactor coolant, α is not so small. In these conditions, Transactions, SMiRT 19, Toronto, August 2007 Paper # K13/1 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 Modified acceleration Acceleration method method method Minimum Maximum Minimum Maximum Minimum Maximum Horizontal displacement -15.06 16.79 -12.81 16.35 -15.45 16.81 at steam nozzle / mm Torsional -15.45 6.31 -15.21 5.78 -15.51 6.25 Elbow at the /104N.m SG inlet Bending moment -14.07 7.01 -14.38 7.06 -13.37 6.98 /104N.m Torsional moment -11.27 7.16 -9.76 6.48 -12.57 8.37 Elbow at the /104N.m SG outlet Bending moment -46.46 26.79 -49.20 28.23 -46.50 33.25 /104N.m Axial force -65.78 67.02 -67.75 72.01 -67.97 68.81 /104N Lateral force -11.47 19.64 -10.57 22.29 -9.95 18.91 RPV Outlet /104N nozzle Torsional moment -15.18 6.14 -14.42 5.84 -15.17 6.01 /104N.m Bending moment -34.15 38.62 -32.75 43.50 -31.50 43.53 /104N.m Horizontal loads -73.35 90.97 -74.22 104.90 -75.49 93.67 Support at /104N RPV inlet Vertical loads -44.22 47.20 -41.57 48.92 -52.43 44.42 /104N Horizontal loads -68.26 75.49 -69.47 73.49 -70.93 78.62 Support at /104N RPV outlet Vertical loads -41.64 42.53 -36.68 50.31 -45.67 47.96 /104N

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 ”, 1973. [4] Hong Jing-fen. “Overview of Seismic Analysis for Nuclear Power Plant,” Nuclear Power Engineering, 17 (3), 1996, pp. 193-198. (in Chinese)