Equivalent Linear Analysis Model With Refined Modeling of Tube and Tube Support Plate Contact for Seismic Response Evaluation of Steam Generator Tube Bundles

Ensuring the structural integrity of Steam Generator (SG) U-bend tube arrays during out-of-plane earthquakes is crucial. Rationalization of the design code is desired considering the increasing trend of design earthquake level. As a rationalization evaluation method, a primary stress criterion of tube arrays obtained from a limit analysis using a nonlinear model which can precisely consider the load sharing mechanism of the tubes by modeling all tubes and anti-vibration-bars (AVBs) is under consideration, and a response analysis is conducted using the equivalent linear model of the nonlinear model and response spectrum method. In this study, a response analysis method which can simultaneously evaluate the response displacement and bending strain distribution using an equivalent linear model and bending strain correction coefficient was proposed. The concept of the proposed method is that the macroscopic response (displacement and strain distributions) of the tube bundle is obtained by the response spectrum method using the equivalent linear model. The nonlinear behavior in which the strain per deformation just above the upper tube support plate (TSP) increases due to the change in the contact state between the tube and the TSP, which cannot be evaluated by the linear model directly, is corrected using the bending strain correction coefficient calculated based on the single tube analysis. The proposed method is validated by comparing the seismic test results obtained based on a full-scale mock-up consisting of square arrayed SG tubes. From the seismic test and pseudo static analysis using the nonlinear model, it was confirmed that near upper tube support plate (TSP) where high stress occurs as the out-of-plane response of the tube bundle increases, a change in contact occurs, resulting in the tube bundle only in contact with the upper surface of the TSP to also enter into contact with the lower surface of the TSP, and that the strain per deformation increases nonlinearly. In the linear model constructed in the previous study, the linear spring is defined at the contact point between the AVBs and the tube, whereas the non-contact point is free. In order to accurately predict stress, an evaluation method considering the contact state at the TSP is required. Therefore, a linear spring was defined only at the contact point at the moment of linearization at the top and bottom contact surfaces of the TSP, and the stiffness of the spring thereof was calculated as the equivalent stiffness from the contact force and gap displacement at the moment of linearization. This equivalent stiffness was also set for the AVB-tube contact point. Theoretically, responses such as deformed shape or strain distribution at the linearization moment, pseudo static analysis of the equivalent linear model and original nonlinear model are consistent. Furthermore, the bending strain correction coefficient, which can be calculated only by nonlinear analysis of a single tube, is set so that the response evaluation becomes conservative. The natural frequency of out-of-plane first mode obtained by the eigenvalue analysis of the equivalent linear model were well consistent with the seismic test. Bending strains of out-of-plane first mode obtained by the response spectrum method using the typical design damping ratio of 8% were larger than the test results. Furthermore, the response displacement was also well consistent with the measured values. Therefore, the proposed method can simultaneously evaluate the displacement response and the bending strain distribution by the response spectrum method. In addition, the natural frequencies of the out-of-plane second-mode were well consistent with the measured values in the linear model which is applied to the evaluation of the out-of-plane second or more higher mode. In conclusion, it was confirmed that the proposed method is applicable as a seismic response evaluation method that includes safety margins.