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Research on Time-History Input Methodology of Seismic Analysis

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Research on Time-History Input Methodology of Seismic Analysis Transactions, SMiRT 19, Toronto, August 2007 Paper # K13/1 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: Transactions, SMiRT 19, Toronto, August 2007 Paper # K13/1 ⎡MMAA AB ⎤⎧X&& ⎫ ⎡CCAA AB ⎤⎧X& ⎫ ⎡KKAA AB ⎤⎧X⎫ ⎧ 0 ⎫ ⎢ T ⎥⎨ ⎬ + ⎢ T ⎥⎨ ⎬ + ⎢ T ⎥⎨ ⎬ = ⎨ ⎬ (1) ⎣MMAB BB ⎦⎩Z&& ⎭ ⎣CCAB BB ⎦⎩Z& ⎭ ⎣KKAB BB ⎦⎩Z⎭ ⎩FB ⎭ 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 K denote 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: MXCXKXMZCZKZAA &&+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, 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= d+ s (3a) ZZZ= d+ s (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 XKKZs = − AA AB (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: −1 −1 MXCXKXMMKKZCCKKZAA&& d +AA & d +AA d =[ AB − AA AA AB ][&& +AB − AA AA AB ] & (5) −1 where MAB is identically equal to a null matrix for a lumped mass formulation; − KKAA AB can be replaced by the matrix R. Thus, Eq. (5) can be simplified as: MXCXKXAA&& d +AA & d +AA d = − MRZCAA && +[ AB − CRZAA ] & (6) The second 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: MAA X&& d + CAA X & d + KAA X d = − MAA RZ&& (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.: CMKAA = α AA + β AA (8a) CMKAB = α AB + β AB (8b) Thus, Eq. (6) may be rewritten as follows: MAA X&& d + CAA X & d + KAA X d = − MAA RZ&& E (9a) ZZZ&&E = && +α & (9b) For modified acceleration method, the velocity, 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 Force / 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.
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