Lumped Mass Stick Model Research Articles

Seismic Fragility Assessment of Equipment in Nuclear Power Plant Structures Considering Nonlinear Hysteretic Behavior of Squat Walls

Accurate seismic risk assessment of structures necessitates precise fragility analysis. This study focuses on the seismic fragility assessment of equipment in nuclear power plant structures, particularly emphasizing the influence of nonlinear hysteretic behavior exhibited by squat walls. By developing and comparing linear and nonlinear models of reactor containment and auxiliary buildings within a nuclear power plant, this research elucidates the significant impact of nonlinear behaviors on the seismic response and subsequent fragility curves of associated equipment and systems. Utilizing a lumped-mass stick model, the study efficiently captures the characteristics of squat shear walls, facilitating the calculation of floor response spectra and fragility for nuclear facility structures. The impacts of pinching and degradation, inherent characteristics of a squat wall’s hysteresis under cyclic loading, were identified as significant. These include the shift and change in amplitude of the floor response’s peak, along with amplification in high-frequency domains. Fragility assessments, informed by seismic response analysis, indicate significant variations in fragility curve parameters based on equipment location and frequency. The findings question the effectiveness of traditional response factor methods and general regression techniques in accurately addressing the complex probabilistic seismic demands of equipment, thus highlighting the importance of using direct analysis for calculating fragility in situations with significant nonlinearity.

Effect of slab stiffness on floor response spectrum and fragility of equipment in nuclear power plant building

The floor response spectrum (FRS) is used to evaluate the seismic demand of equipment installed in nuclear power plants. In the conventional design practice of NPP structure, the FRS is simplified using the lumped-mass stick model (LMSM), assuming the floor slab as a rigid diaphragm. In the present study, to study the variation of seismic response in a floor, the FRSs at different locations were generated by 3-D finite element model, and the response was compared to that of the rigid diaphragm model. The result showed that the FRS significantly varied due to the large opening in a floor, which was not captured by the rigid diaphragm model. Based on the result, seismic fragility analysis was performed for the anchorage of a heat exchanger, to investigate the effect of location-dependent FRS disparity on the high confidence low probability of failure (HCLPF).

Correlation Analysis of Earthquake Intensity Measures and Engineering Demand Parameters of Reactor Containment Structure

This study aims to analyze the correlation between earthquake Intensity Measures (IMs) and seismic responses of a reactor containment building in an APR-1400 nuclear power plant. A total of 20 IMs were employed to develop Seismic Demand Regression Models (SDRMs), which show the relationship between IMs and engineering demand parameters. A numerical model of the structure was constructed using the Lumped-Mass Stick Model (LMSM) in SAP2000. Additionally, a three-dimensional finite element model was developed to validate the simplified LMSM approach. A set of 90 ground motion records was used to perform a time-history analysis, where the motions cover a wide range of amplitude, intensity, epicenter distance, significant duration, and frequency of earthquakes. Engineering demand parameters were monitored in terms of floor accelerations and displacements. Consequently, strongly correlated IMs were identified based on the evaluation of SDRMs using four statistical indicators: coefficient of determination, standard deviation, practicality, and proficiency. The results showed that the strongest IMs were Sa(T1), Sv(T1), and Sd(T1) followed by ASI, EPA, PGA, and A95. On the other hand, the weakly correlated IMs were PGD, DRMS, SED, VRMS, PGV, HI, VSI, and SMV.

Optimal Earthquake Intensity Measures for Probabilistic Seismic Demand Models of Base-Isolated Nuclear Power Plant Structures

The purpose of this study is to evaluate the optimal earthquake intensity measures (IMs) for probabilistic seismic demand models (PSDMs) of the base-isolated nuclear power plant (NPP) structures. The numerical model of NPP structures is developed using a lumped-mass stick model, in which a bilinear model is employed to simulate the force-displacement relations of base isolators. In this study, 20 different IMs are considered and 90 ground motion records are used to perform time-history analyses. The seismic engineering demand parameters (EDPs) are monitored in terms of maximum floor displacement (MFD), the maximum floor acceleration (MFA) of the structures, and maximum isolator displacement (MID). As a result, a set of PSDMs of the base-isolated structure is developed based on three EDPs (i.e., MFD, MFA, and MID) associated with 20 IMs. Four statistical parameters including the coefficient of determination, efficiency (i.e., standard deviation), practicality, and proficiency are then calculated to evaluate optimal IMs for seismic performances of the isolated NPP structures. The results reveal that the optimal IMs for PSDMs with respect to MFD and MID are velocity spectrum intensity, Housner intensity, peak ground velocity, and spectral velocity at the fundamental period. Meanwhile, peak ground acceleration, acceleration spectrum intensity, A95, effective peak acceleration, and sustained maximum acceleration are efficient IMs for PSDMs with respect to MFA of the base-isolated structures. On the other hand, cumulative absolute velocity is not recommended for determining the exceedance of the operating basis earthquake of base-isolated NPP structures.

Seismic performance assessment of NPP concrete containments considering recent ground motions in South Korea

Seismic fragility analysis, a part of seismic probabilistic risk assessment (SPRA), is commonly used to establish the relationship between a representative property of earthquakes and the failure probability of a structure, component, or system. Current guidelines on the SPRA of nuclear power plants (NPPs) used worldwide mainly reflect the earthquake characteristics of the western United States. However, different earthquake characteristics may have a significant impact on the seismic fragility of a structure. Given the concern, this study aimed to investigate the effects of earthquake characteristics on the seismic fragility of concrete containments housing the OPR-1000 reactor. Earthquake time histories were created from 30 ground motions (including those of the 2016 Gyeongju earthquake) by spectral matching to the site-specific response spectrum of Hanbit nuclear power plants in South Korea. Fragility curves of the containment structure were determined under the linear response history analysis using a lumped-mass stick model and 30 ground motions, and were compared in terms of earthquake characteristics. The results showed that the median capacity and high confidence of low probability of failure (HCLPF) tended to highly depend on the sustained maximum acceleration (SMA), and increase when using the time histories which have lower SMA compared with the others.

A Short Review on Numerical Modelling Approaches for Seismic Evaluation Performance of Nuclear Power Plant Structures

Nuclear power plant (NPP) structures play a crucial role in protecting the safety of the whole plant. Overall, NPP structures have complex shapes and large dimensions. Therefore, the decision of an appropriate finite element model for seismic response analysis is important. This study presents a brief review of various numerical modelling approaches for seismic evaluation performance of NPP structures. Different conventional models, i.e. lumped-mass stick model (LMSM), full three-dimensional finite element model (3D FEM), elastic solid element model (ESM), and multi-layer shell model (MLSM), which have been applied in modelling nuclear structures, are introduced. Also, the advantages and drawbacks of those models are analysed. Furthermore, a new model namely, beam-truss model (BTM), which is recently proposed, is highlighted. It reveals that LMSM is the most simplified approach for structural modelling of NPP structures. However, it is normally used for linear analyses and not able to simulate the local behaviours and vertical responses of the complex NPP structures. Even though 3D FEM is the most sufficient method for nonlinear seismic response analyses, this approach is very time-consuming and costly computation. MLSM and BTM are recommended as practical and efficient models for nonlinear analyses of NPP structures.

Efficiency of various structural modeling schemes on evaluating seismic performance and fragility of APR1400 containment building

The purpose of this study is to investigate the efficiency of various structural modeling schemes for evaluating seismic performances and fragility of the reactor containment building (RCB) structure in the advanced power reactor 1400 (APR1400) nuclear power plant (NPP). Four structural modeling schemes, i.e. lumped-mass stick model (LMSM), solid-based finite element model (Solid FEM), multi-layer shell model (MLSM), and beam-truss model (BTM), are developed to simulate the seismic behaviors of the containment structure. A full three-dimensional finite element model (full 3D FEM) is additionally constructed to verify the previous numerical models. A set of input ground motions with response spectra matching to the US NRC 1.60 design spectrum is generated to perform linear and nonlinear time-history analyses. Floor response spectra (FRS) and floor displacements are obtained at the different elevations of the structure since they are critical outputs for evaluating the seismic vulnerability of RCB and secondary components. The results show that the difference in seismic responses between linear and nonlinear analyses gets larger as an earthquake intensity increases. It is observed that the linear analysis underestimates floor displacements while it overestimates floor accelerations. Moreover, a systematic assessment of the capability and efficiency of each structural model is presented thoroughly. MLSM can be an alternative approach to a full 3D FEM, which is complicated in modeling and extremely time-consuming in dynamic analyses. Specifically, BTM is recommended as the optimal model for evaluating the nonlinear seismic performance of NPP structures. Thereafter, linear and nonlinear BTM are employed in a series of time-history analyses to develop fragility curves of RCB for different damage states. It is shown that the linear analysis underestimates the probability of damage of RCB at a given earthquake intensity when compared to the nonlinear analysis. The nonlinear analysis approach is highly suggested for assessing the vulnerability of NPP structures.

Comparison of Seismic Responses of Updated Lumped-Mass Stick Model and Shaking Table Test Results

A conventional lumped-mass stick model is based on the tributary area method to determine the masses lumped at each node and used in earthquake engineering due to its simplicity in the modeling of structures. However the natural frequencies of the conventional model are normally not identical to tho...

Development of basic technique to improve seismic response accuracy of tributary area-based lumped-mass stick models

Although a detailed finite element (FE) model provides more precise results, a lumped-mass stick (LMS) model is preferred because of its simplicity and rapid computational time. However, the reliability of LMS models has been questioned especially for structures dominated by higher modes and seismic inputs. Normally, the natural frequencies and dynamic responses of a LMS model based on tributary area mass consideration are different from the results of the FE model. This study proposes a basic updating technique to overcome these discrepancies; the technique employs the identical modal response, D(t), to the detailed FE model. The parameter D(t) is a time variable function in the dynamic response composition and it depends on frequency and damping ratio for each mode, independent of the structure’s mode shapes. The identical response D(t) for each mode is obtained from the frequency adaptive LMS model; the adaptive LMS model which can provide identical modal frequencies as the detailed FE model. Theoretical backgrounds and formulations of the updating technique are proposed. To validate the updating technique, two types of structures (a symmetric straight column and an unsymmetric T-shaped structure) are considered. From the seismic response results including base shear and base moment, the updating technique considerably improves the seismic response accuracy of the tributary area-based LMS model.

Wavelet analysis of soil-structure interaction effects on seismic responses of base-isolated nuclear power plants

Seismic base isolation has been accepted as one of the most popular design procedures to protect important structures against earthquakes. However, due to lack of information and experimental data the application of base isolation is quite limited to nuclear power plant (NPP) industry. Moreover, the effects of inelastic behavior of soil beneath base-isolated NPP have raised questions to the effectiveness of isolation device. This study applies the wavelet analysis to investigate the effects of soil-structure interaction (SSI) on the seismic response of a base-isolated NPP structure. To evaluate the SSI effects, the NPP structure is modelled as a lumped mass stick model and combined with a soil model using the concept of cone models. The lead rubber bearing (LRB) base isolator is used to adopt the base isolation system. The shear wave velocity of soil is varied to reflect the real rock site conditions of structure. The comparison between seismic performance of isolated structure and non-isolated structure has drawn. The results show that the wavelet analysis proves to be an efficient tool to evaluate the SSI effects on the seismic response of base-isolated structure and the seismic performance of base-isolated NPP is not sensitive to the effects in this case.

Effect of Soil-Structure Interaction on the Seismic Fragility of a Nuclear Reactor Building

Recently, the nuclear industry has made a tremendous effort to assess the safety of nuclear power plants (NPP), as advances in seismology have led to the perception that the potential earthquake hazard in the U.S. may be higher than originally assumed. Due to the conservatism in the NPP design, structures and safety-related items are capable of withstanding earthquakes larger than the safe shutdown earthquake (SSE). One major aspect of conservatism in the design is ignoring the effect of soil-structure interaction (SSI), which results in conservative estimates of seismic demands for plant equipment. In this paper, a typical reactor building (RB) is chosen for a case study to investigate the potential benefit of accounting for SSI effects. A lumped mass stick model is first developed and analyzed with a fixed base configuration, using the free-field ground motion as input at the foundation level, as well as with a SSI configuration. Fragility analyses are then performed for the RB and one of its components to quantify the effects of the SSI on the overall seismic risk. In each case, a family of seismic fragility curves is developed. It is found that consideration of SSI effects in the analysis can improve the component fragilities, and potentially enhance the core damage frequency (CDF) of the plant.

全体曲げ変形に伴う制振建物の性能低下を補償する同調質量ダンパーの最適設計法

It is being revealed that occurrence of the Nankai-Trough earthquake might cause significant damage to low-damped super high rise steel buildings in some metropolitan areas in Japan. Addition of supplemental damping devices is effective means to mitigate the damage caused by the great earthquake. However, the seismic effectiveness of dampers connected to a structural frame tends to lose due to overall bending deformation in a slender steel building. On the other hand, a tuned mass damper (TMD) connected to the frame work well even if the overall bending deformation is quite large. The combination of the two kinds of dampers therefore is expected to be effective in slender buildings with a large aspect ratio. However, an optimal design parameter of the TMD mounted on highly damped buildings and its seismic effectiveness are not clarified. The objective of this paper is to develop a design procedure of the two different damping devices. Both the dampers are limited to a linear viscous damper in this study. Firstly, a lumped mass stick model with shear springs and rotational springs is employed to take account of the overall bending deformation. Two degree of freedom (2-DOF) system is derived from this model with the help of dynamic reduction technique. The 2-DOF system becomes non-proportional damping system. Subsequently, a linear viscous TMD are connected to the 2-DOF system and a 3-DOF system is obtained for the TMD-building interaction system. Optimal design parameters of the TMD are obtained by solving a Lyapunov equation of the 3-DOF system. The optimal stiffness of the TMD is summarized in terms of equivalent damping factor due to only dampers installed into the stories. The computed optimal stiffness is almost consistent with the result of SDOF system with the same equivalent damping factor. This fact admits to use regression formulae of the optimal stiffness proposed by several researchers. It is also confirmed that the optimal parameters obtained by the criterion of minimizing top displacement and story drift is almost the same. Secondary, equivalent damping factor of the TMD-building interaction system are investigated through stochastic dynamic analysis. As a result, the equivalent damping decreases according to increase of the amount of the dampers within stories. The convenient regression formula is proposed in terms of the equivalent damping factor of the building. The equivalent damping factor is directly evaluated with a slenderness ratio of the building and a damping factor by using the formula. Finally, a design procedure of the TMD is formulated. Required amount of the TMD mass is proposed based on the concept that the loss of damping factor is equal to the additional equivalent damping factor added by the TMD.

Seismic vulnerability mitigation of liquefied gas tanks using concave sliding bearings

The aim of this paper is to evaluate the effectiveness of a concave sliding bearing system for the seismic protection of liquefied gas storage tanks through a seismic fragility analysis. An emblematic case study of elevated steel storage tanks, which collapsed during the 1999 Izmit earthquake at Habas Pharmaceutics plant in Turkey, is studied. Firstly, a fragility analysis is conducted for the examined tank based on a lumped-mass stick model, where the nonlinear shear behaviour of support columns is taken into account by using a phenomenological model. Fragility curves in terms of an efficient intensity measure for different failure modes of structural components demonstrate the inevitable collapse of the tank mainly due to insufficient shear strength of the support columns. A seismic isolation system based on concave sliding bearings, which has been demonstrated a superior solution to seismically protect elevated tanks, is then designed and introduced into the numerical model, accounting for its non-linear behaviour. Finally, a vulnerability analysis for the isolated tank is performed, which proves a high effectiveness of the isolation system in reducing the probability of failure within an expected range of earthquake intensity levels.

Dynamic response of cylindrical structures considering coupled soil-structure interaction under seismic loading

This paper attempts to develop a mathematical model for estimating the seismic response of a cylindrical shaped nuclear reactor building resting in an elastic halfspace considering foundation compliance. Most of the research carried out on this topic has either been carried out by resorting to finite element method (FEM) which makes the computational cost expensive or based on the simplifying assumption of assuming the cylindrical structure as a multi degree lumped mass stick model with soil coupled as boundary springs. In the present paper an analytical model has been developed in which the deformation of the cylindrical body (including its shear deformation characteristics) has been taken into cognizance and then coupling with foundation stiffness a comprehensive solution has been sought based on Galerkin’s weighted residual technique. The results are finally compared with FEM to check the reliability of the same. The results are found to be in good agreement with the detailed finite element analysis.

Effect of in-situ variability of soil on seismic design of piled raft supported structure incorporating dynamic soil-structure-interaction

Inherent variability of soil considerably affects the seismic design of piled raft supported structures. Conventional design of such structure adopts fixity at base level of superstructure and pile head. However, soil–pile–superstructure interaction largely affects the fundamental frequency and design forces in columns and piles. In contrast, fixed base assumption cannot capture soil structure interaction (SSI) effect. In addition, uncertainty in soil may further leads to a change in the dynamic behavior of the system. This study examines the effect of inherent variability of undrained shear strength of soil in seismic design of structures supported by piled raft foundation embedded in soft clay. Superstructure is modeled as lumped mass stick model and piled raft slab is modeled as rigid plate. Pile is modeled as Euler–Bernoulli beam element and soil resistance is modeled using linear Winkler springs attached to the pile. Dynamic analysis is carried out in time domain to estimate the responses. Monte Carlo simulation technique is used for probabilistic assessment of the fundamental frequency and forces at column and pile attributing a wide range of parametric variation of a representative soil–piled raft–superstructure system. The study shows that the fundamental frequency and forces in column and pile changes significantly due to soil variability.

Seismic Performance of High Strength Reinforced Concrete Buildings Evaluated by Nonlinear Pushover and Dynamic Analyses

This paper investigates the combined effect of flexural and shear actions on the failure modes of the high strength reinforced concrete (HRC) members using the proposed algorithm for plastic hinge formation. The accuracy of the present procedure for the HRC columns was verified by comparing the results obtained with those of the cyclic loading tests performed in Japan. To evaluate the seismic performance of the HRC high-rise buildings, a seismic performance checklist for the HRC buildings was recommended. Based on the proposed algorithm for formation of plastic hinges, the seismic performance of HRC buildings based on the static pushover analysis is evaluated. From the results of the pushover analysis, a simplified lumped-mass stick model was developed, which is adopted to evaluate the seismic performance using the nonlinear time history analysis. For the purpose of illustration, the seismic performance of a high-rise building constructed with HRC was investigated by both the nonlinear pushover and nonlinear dynamic analyses using the proposed procedure and concepts. The results of this paper serve as a useful reference for the seismic design and evaluation of HRC high-rise structures.

New lumped-mass-stick model based on modal characteristics of structures: development and application to a nuclear containment building

In this study, a new lumped-mass-stick model (LMSM) is developed based on the modal characteristics of a structure such as eigenvalues and eigenvectors. The simplified model, named the “frequency adaptive lumped-massstick model,” hasonly a small number of stick elements and nodes to provide the same natural frequencies of the structure and is applied to a nuclear containment building. To investigate the numerical performance of the LMSM, a time history analysis is carried out on both the LMSM and the finite element model (FEM) for a nuclear containment building. A comparison of the results shows that the dynamic responses of the LMSM in terms of displacement and acceleration are almost identical to those of the FEM. In addition, the results in terms of fl oor response spectra at certain elevations are also in good agreement.

Response Spectrum Analysis of the Condensate Storage Tank in a Nuclear Power Plant

Following the nuclear power plant accident in Fukushima Japan, seismic capacity evaluation has become a crucial issue in combination building safety. Condensate storage tanks are designed to supplies water to the condensate transfer pumps, the control rod drive hydraulic system pumps, and the condenser makeup. A separate connection to the condensate storage tank is used to supply water for the high pressure coolant injection system, reactor core isolation cooling system, and core spray system pumps. A condensate storage tank is defined as a seismic class I structure, playing the important role of providing flow to the operational system and the required static head for the suction of the condensate transfer pumps and the normal supply pump. According to the latest nuclear safety requirements, soil structure interaction must be considered in all seismic analyses. This study aims to rebuild the computer model of condensate storage tanks in Taiwan using the SAP 2000 program in conjunction with the lumped mass stick model and to evaluate the soil structure interaction by employing the SASSI 2000 program. The differences between the results with the soil structure interaction and spring model are compared via natural frequency and response spectrum curves. This computer model enables engineers to rapidly evaluate the safety margin of condensate storage tank following the occurrence of earthquakes or tsunamis.

구조물의 동적 고유특성을 이용한 새로운 집중질량모델 개발

구조물의 내진설계 또는 내진성능평가를 위해서는 구조물의 축소모형을 이용한 실험적 분석이나 유한요소모델을 기반으로 한 수치적 방법이 고려된다. 수치적 방법을 위해서는 정교한 모델링이 요구될 경우 3차원 유한요소해석을 실시하나 민감도 분석이나 지진 취약도 분석과 같은 방대한 지진데이터를 이용한 평가에서는 집중질량모델이 선호된다. 하지만 기존의 집중질량모델은 일반적으로 구조물의 기하학적 형상을 고려하여 집중질량을 산출하는 방식인데, 이 경우 제공되는 고유치는 실구조물의 고유치와 일치하지 않는다. 본 연구에서는 이러한 문제점을 개선하고 실구조물과 유사한 동적 거동을 발현하는 새로운 형식의 주파수 순응형 집중질량모델을 제안하였다. 제안된 모델은 실구조물의 고유치와 고유 벡터, 모드 형상 등을 고려하여 생성하며, 모델의 성능을 검증하기 위해 비균일 단면을 갖는 기둥에 대해 동적해석을 수행하였다. 또한 감쇠비에 따른 동적성능을 분석하기 위해 1%에서 5%까지의 Rayleigh Damping 적용하여 그 결과를 유한요소모델 결과와 비교하였다. For a seismic design or performance evaluation of a structure, an experimental investigation on a scale model of the structure or numerical analysis based on the finite element model is considered. Regarding the numerical analysis, a three-dimensional finite element analysis is performed if a high accuracy of the results is required, while a sensitivity or fragility analysis which uses huge seismic ground motions leads to the use of a lumped-mass stick model. The conventional modeling technique to build the lumped-mass stick model calculates the amount of the lumped mass by considering the geometric shape of the structure, like a tributary area. However, the eigenvalues of the conventional model obtained through such a calculation are normally not the same as those of the actual structure. In order to overcome such a deficiency, in this study, a new lumped mass stick model is proposed. The model is named the "frequency adaptive-lumped-mass stick model." It provides the same eigenvalues and similar dynamic responses as the actual structure. A non-prismatic column is considered as an example, and its natural frequencies as well as the dynamic performance of the new lumped model are compared to those of the full-finite element model. To investigate the damping effect on the new model, 1% to 5% of the critical damping ratio is applied to the model and the corresponding results are also compared to those of the finite element model.

Rebuilding the Combination Building of a Nuclear Power Plant by Employing the Lumped Mass Stick Model

This study rebuilds the computer model of combination building for preparations to enhance the seismic design of nuclear power plants. The lumped mass stick model is used to generate the natural frequency and floor response spectrum, and this paper conducts a comparison of the variations between the original and rebuilt model before an evaluation. The analysis results indicate that the natural frequency and floor response spectrum provided by the novel lumped mass stick model represent the overall building responses in a reasonable fashion. The results generated by the lumped mass model fit those generated by the previous calculations in 1986. The lumped mass model is suitable for performing the safe evaluation of combination building after an occurrence of earthquakes and tsunamis.