A new algebraic white-noise modal combination rule — GAC(A)

It is a well known fact that above the rigid frequency the maximum dynamic modal responses even with different multiple supports can be combined algebraically. Below the rigid frequency, and more specifically in the white-noise (amplified) region of the response spectrum, algebraic modal combination is still a matter of controversy, demonstrated e.g. by the NRC R.G. 1.92 requirement to take the absolute values of the modal responses in heuristic modal combination rules; whereas algebraic support combination is only allowed in conjunction with the envelope support response spectrum (ERS). Such regulatory requirements can lead to unrealistically high calculated responses e.g. in the coupled analysis of light secondary systems attached to heavy primary structures and in the decoupled analysis of systems when the centers of mass and stiffness do not coincide, or when the ERS is used.A new Generalised Algebraic Combination (GAC) methodology has been theoretically developed which allows practical algebraic modal and support combination over the whole frequency range of multiple support spectra. The present paper deals with with the GAC-(A) i.e. the modal combination version in the white noise region of a single response spectrum and more specifically its time history integration validation, which shows that this new modal combination rule can satisfy any realistic conservatism that may be required by regulatory institutions.     

Determination of earthquake loads on structures — Research and development required

This paper discusses the research and development work required in three aspects of seismic analysis, i.e. modal combination criteria in the response spectrum analysis, the techniques in modeling and simulating structures, systems and components, and the determination of dynamic lateral earth pressure during an earthquake.In the seismic analysis of structures, the lumped mass method is commonly used. For this method, the thin wall beam theory approach and the rigidity approach used in determining the effective shear areas and the distributions of translational shears are discussed. In the analysis of structures, either the time history or response spectrum method is used. When the response spectrum modal analysis method is used, the exact phase angle relation among maximum modal responses is not defined; therefore, a logical combination criterion must be established. The criterion for closely spaced modal responses to obtain system response is also mentioned. Finally, the significantly different lateral earth pressures on basement walls of embedded structures obtained using the Mononobe-Okabe method and the finite element method are also discussed. Hopefully, through further research and development work in these areas, safe and economic design can be achieved.     

Comparison of multiple support excitation solution techniques for piping systems

Design and analysis of nuclear power plant piping systems exposed to a variety of dynamic loads often require multiple support excitation analysis by modal or direct time integration methods. Both methods have recently been implemented in the computer program KWUROHR for static and dynamic analysis of piping systems, following the previous implementation of the multiple support excitation response spectrum method (see papers K6/15 and K6/15a of the SMiRT-4 Conference).The results of multiple support excitation response spectrum analyses can be examined by carrying out the equivalent time history analyses which do not distort the time phase relationship between the excitations at different support points.A frequent point of discussion is multiple versus single support excitation. A single support excitation analysis is computationally straightforward and tends to be on the conservative side, as the numerical results show. A multiple support excitation analysis, however, does not incur much more additional computer cost than the expenditure for an initial static solution involving three times the number, L, of excitation levels, i.e. 3L static load cases. The results are more realistic than those from a single support excitation analysis.A number of typical nuclear plant piping systems have been analyzed using single and multiple support excitation algorithms for: (1) the response spectrum method, (2) the modal time history method via the Wilson, Newmark and Goldberg integration operators and (3) the direct time history method via the Wilson integration operator. Characteristic results are presented to compare the computational quality of all three methods.     

Comparison of modal combination methods

A new method of combining modal responses in the response spectrum method of analysis was recently presented by Gupta and Cordero, which is designated here as Gupta's method. It is shown that the response spectrum method in conjunction with Gupta's method of modal combination gives results which are very close to those obtained from the time-history analysis. Further, Gupta's method gives results which are more accurate than other methods of modal combination discussed here, viz., SRSS, Kennedy's and Hadjian's.In Gupta's method response in any mode is split into two parts, the damped periodic part and the rigid part, which are mutually uncorrelated. This is done using a rigid response factor. A simple expression for the rigid response coefficient is presented. The key frequencies in the expression can be evaluated from the motion response spectrum. There is good agreement between the proposed expression and the numerically obtained rigid response coefficient.     

Combination of modal responses: A closed-form formulation for rigid response coefficient

Recent studies conducted by US Nuclear Regulatory Commission (USNRC) for combining modal responses in a response spectrum method of seismic analysis and design have emphasized that each modal response quantity should be separated into damped-periodic and rigid parts before combining the contributions from different modes. The damped-periodic parts of modal responses are combined using the double-sum equation whereas the rigid parts are combined algebraically. A particular modal response quantity is separated into damped-periodic and rigid parts using the “rigid response coefficient”. The USNRC sponsored study recommends the calculation of rigid response coefficient by either the Lindley–Yow approach or Gupta method. While Lindley–Yow's method has a heuristic basis and gives incorrect results in low frequency region, Gupta's method is based on numerical studies of free-field earthquake motions and works well in the frequency regions of interest for a free-field ground motion. A closed-form solution was developed by Hahn and Valenti in 1997 using a frequency domain approach. With appropriate simplifications, their work can be shown to result in an expression which is very similar to that given by Gupta. It must be noted that the earthquake input to the secondary systems such as piping and equipment is defined by a floor motion and not a free-field ground motion. The frequency characteristics of a floor motion are very different from those of a free-field ground motion. In this paper, we study the validity of existing formulations for the case of floor motions and develop a closed-form solution based on a time domain approach to explain the behavior of rigid response coefficient. The formulation is then used to explain the nature of variation in rigid response coefficient for ground as well as floor motions. It is shown that the proposed formulation and its simplified form gives results that are identical to those evaluated numerically in the complete frequency region of interest.     

Response spectrum analysis of piping systems with elastic-plastic gap supports

A new seismic support device and its application in piping systems is described. The device, E-BAR (patented), can be cost effectively used for snubber replacement programs, mitigation of hydraulic transients, pipe whip and as a thermal stop. The device has pre-set gaps to allow free thermal movement. During a seismic or other dynamic load event, if the pipe movement exceeds the gap dimension, the device acts as an elastic or elastic-plastic restraint. The device also has a unique design feature for not exceeding the restraint force beyond a specified limit design value. To analyze piping systems with gap supports having elastic-plastic characteristics, modal analysis procedures for both response spectrum and time history methods are developed. The comparison of responses obtained from the procedures with nonlinear time history analysis and test results available in the literature shows excellent correlation. A pilot program conducted for snubber replacement with E-BARs demonstrates that the limit force feature of E-BAR makes them very attractive for snubber replacement. This is because a particular E-BAR with a specified limit design force can be selected, such that, the E-BAR replacing the snubber does not require any modifications be made to the existing support steel and hardware.     

A new instructure response spectrum (IRS) method for multiply connected secondary systems with coupling effects

A method of performing coupled response spectrum analysis of secondary systems is presented. The response spectrum specified at the base of the primary system is used as the input. The complex coupled mode shapes along with frequencies and damping values are calculated using an efficient and accurate perturbation scheme. The new method is applied to a 2 DOF secondary system coupled with a 6 DOF secondary system. The masses and the stiffness of the secondary system are varied to get nine different cases. The coupled system is subjected to El Centro (S00E, 1940) ground motion. It is shown that the response values from the present method are in good agreement with those from the coupled time history analysis. The conventional floor response spectrum method gives response values which are consistently much higher than the corresponding values from the time history analysis. It is concluded that the present method is sufficiently straightforward and efficient, and that it yields accurate response values.     

Seismic response analysis of structural system subjected to multiple support excitation

In the seismic analysis of a multiply supported structural system subjected to nonuniform excitations at each support point, the single response spectrum, the time history, and the multiple response spectrum are the three commonly employed methods. In the present paper the three methods are developed, evaluated, and the limitations and advantages of each method assessed. A numerical example has been carried out for a typical piping system. Considerably smaller responses have been predicted by the time history method than that by the single response spectrum method. This is mainly due to the fact that the phase and amplitude relations between the support excitations are faithfully retained in the time history method. The multiple response spectrum prediction has been observed to compare favorably with the time history method prediction. Based on the present evaluation, the multiple response spectrum method is the most efficient method for seismic response analysis of structural systems subjected to multiple support excitation.     

An improved spectrum superposition method for structures with rigid modes

Based on a precise probabilistic formulation, a spectrum superposition method has been proposed to obtain very accurate estimate of the seismic response of structures with significant contribution from high-frequency rigid modes. The proposed approach computes the exact statistics of the ordered-peaks of response, which is defined in terms of the first few moments of the power spectral density function (PSDF) of the response. Like many other studies, the PSDFs for various response quantities have been defined in terms of the PSDF of input ground acceleration, the modal properties of a few lowest frequency modes and the pseudostatic response of the structure to an acceleration of unit magnitude. However, the other methods express the final results directly in terms of the response spectrum amplitudes under several simplifying assumptions, which results in unacceptable error in many cases. Due to its exactness, the proposed method has been found to give consistently good agreement with the exact solution for a wide variety of input excitations and structural systems. Furthermore, the present method also has the response spectrum as its basis by defining the PSDF of input excitation which is compatible with a given design spectrum.     

Artificial ground motion compatible with specified peak ground displacement and target multi-damping response spectra

With respect to the design ground motion of nuclear power plant (NPP), the Regular Guide 1.60 of the US not only defined the standard multi-damping response spectra, i.e. the RG1.60 spectra, but also definitely prescribed the peak ground displacement (PGD) value corresponding to the standard spectra. However, in the engineering practice of generating multi-damping-spectra-compatible artificial ground motion for the seismic design of NPP, the PGD value had been neglected. Addressing this issue, this paper proposed a synthesizing method which generates the artificial ground motion compatible with not only the target multi-damping response spectra but also the specified PGD value. Firstly, by the transfer formula between the power spectrum and the response spectrum, an initial uniformly modulated acceleration time history is synthesized by multiplying the stationary Gaussian process with the prescribed intensity envelope to simulate the amplitude-non-stationarity of earthquake ground motion. And then by superimposing a series of narrow-band time histories in the time domain, the initial time history is modified in the iterative manner to match the target PGD as well as the target multi-damping spectra with the pre-specified matching precisions. Numerical examples are provided to demonstrate the matching precisions of the proposed method to the target values.     

Criteria for seismic analysis of nuclear plant structures and substructures

This paper encompasses criteria used for seismic analysis of nuclear power plant structures such as supporting structures founded on ground, as well as substructures. Nuclear power plant equipment and systems can be treated as substructures. Modeling of structures and substructures is described. Since instructure response spectra play an important role in the design and analysis of nuclear power plant equipment, systems and components, methods for development of instructure response spectra as well as variations of input parameters considered in determining these spectra are described.When the principal contribution to the equipment response is due to flexibility of the supporting substructures, an analytical approach to the problem for obtaining reduced stiffness and associated mass matrices of supporting substructures with finite element representation for use in the dynamic analysis of equipment and supporting structures is presented. When supporting structures and equipment, that have inherently different damping properties, are included as intergral parts of the dynamic models, the approximate evaluation of the modal damping based on the weighted damping according to the modal energy stored in each component is outlined. Use of time history and response spectrum analyses is presented. The effects of relative displacements due to different motion of the support points of substructures in each significant mode of the supporting structures as well as procedures of combining modal responses are detailed.     

核电站严重事故后果概率安全评价方法研究

核电站严重事故后果概率安全评价(PSA)是采用概率论的方法对核电站放射性后果进行分析,并定量给出放射性物质对核电站周围公众的健康效应影响。以国内某压水堆核电站为参考厂址,建立合适的场外后果分析模型。采用分层抽样方法对参考厂址1a的气象数据进行抽样,源项和释放特征等数据取自二级PSA的研究结果。利用事故后果评价程序对核电站严重事故后果进行计算,并用概率论方法对结果进行评估。通过计算将各事故和事故谱的场外个人剂量表示为CCDF曲线和总频率-剂量曲线,再用概率论方法得到不同距离处个人剂量超过指定剂量的条件概率;也可用此方法对确定烟羽应急计划区的安全准则中所描述的"大多数严重事故序列"进行量化。     

模态应变能在反应堆及一回路系统动力分析中的应用

为对系统级模型中不同部件和设备动力贡献程度进行量化考察，提出了一种基于模态应变能的计算方法。应用该方法对2个工程案例进行了分析。首先对某堆型反应堆冷却剂系统波动管支吊架位置变更导致的地震响应较大变化的原因进行了分析。分析发现，波动管局部主导模态由于支吊架位置变化而发生变化，主导频率所对应的输入地震响应谱位置相应变化，进而影响了波动管附近的地震响应。然后，本文对同堆型环路模型动力分析中蒸汽发生器主蒸汽管是否能从环路模型中解耦进行了论证。分析发现，根据USNRC SRP 3.7.2解耦准则的第二条，主蒸汽管满足解耦条件，可在动力分析中单独进行处理。本文所提出的分析方法可定量反映部件和设备在系统模型中的动力贡献程度，模态应变能的计算仅应用了系统级模型的质量和刚度信息，无需对整个系统进行时程瞬态分析。      

Soil–structure interaction in fuel handling building

This paper presents an accurate three-dimensional seismic soil–structure interaction analysis for large structures. The method is applied to the fuel building in nuclear power plants. The analysis is performed numerically in the frequency domain and the responses are obtained by inverse Fourier transformation. The size of the structure matrices is reduced by transforming the equation of motion to the modal coordinate system. The soil is simulated as a layered media on top of viscoelastic half space. Soil impedance matrices are calculated from the principles of continuum mechanics and account for soil stiffness and energy dissipation. Effects of embedment on the field equations is incorporated through the scattering matrices or by simply scaling the soil impedance. Finite element methods are used to discretize the concrete foundation for the generation of the soil interaction matrices. Decoupling of the sloshing water in the spent fuel pools and the free-standing spent fuel racks is simulated. The input seismic motions are defined by three artificial time history accelerations. These input motions are generated to match the ground design basis response spectra and the target power spectral density function. The methods described in this paper can handle arbitrary foundation layouts, allows for large structural models, and accurately represents the soil impedance. Time history acceleration responses were subsequently used to generate floor response spectra at applicable damping values.     

A NRC-BNL benchmark evaluation of seismic analysis methods for non-classically damped coupled systems

Under the auspices of the U.S. Nuclear Regulatory Commission (NRC), Brookhaven National Laboratory (BNL) developed a comprehensive program to evaluate state-of-the-art methods and computer programs for seismic analysis of typical coupled nuclear power plant (NPP) systems with non-classical damping. In this program, four benchmark models of coupled building-piping/equipment systems with different damping characteristics were developed and analyzed by BNL for a suite of earthquakes. The BNL analysis was carried out by the Wilson-θ time domain integration method with the system-damping matrix computed using a synthesis formulation as presented in a companion paper [Nucl. Eng. Des. (2002)]. These benchmark problems were subsequently distributed to and analyzed by program participants applying their uniquely developed methods and computer programs. This paper is intended to offer a glimpse at the program, and provide a summary of major findings and principle conclusions with some representative results.The participant’s analysis results established using complex modal time history methods showed good comparison with the BNL solutions, while the analyses produced with either complex-mode response spectrum methods or classical normal-mode response spectrum method, in general, produced more conservative results, when averaged over a suite of earthquakes. However, when coupling due to damping is significant, complex-mode response spectrum methods performed better than the classical normal-mode response spectrum method. Furthermore, as part of the program objectives, a parametric assessment is also presented in this paper, aimed at evaluation of the applicability of various analysis methods to problems with different dynamic characteristics unique to coupled NPP systems. It is believed that the findings and insights learned from this program will be useful in developing new acceptance criteria and providing guidance for future regulatory activities involving license applications of these alternate methods to coupled systems.     

Design and Seismic Analysis of the Large Biological Shielding Door in Nuclear/Fusion Reactor

The large biological shielding door is the important protective equipment of nuclear/fusion reactor. Considering the seismic load and the gravity, it is designed. Based on the modal theory, the modal analysis is performed to obtain the natural frequency and the vibration modes of the door. Using the response spectrum method, the seismic analysis is performed to obtain the stress and displacement. Results show that the biological shielding door can meet the requirement of ASME and the design is reasonable and feasible; the horizontal acceleration being perpendicular to the door leaf is the major factor for the design of the door.     

Lower bound earthquake magnitude for probabilistic seismic hazard evaluation

This paper presents the results of a study that develops an engineering and seismological basis for selecting a lower-bound magnitude (LBM) for use in seismic hazard assessment. As part of a seismic hazard analysis the range of earthquake magnitudes that are included in the assessment of the probability of exceedance of ground motion must be defined. The upper-bound magnitude is established by earth science experts based on their interpretation of the maximum size of earthquakes that can be generated by a seismic source. The lower-bound or smallest earthquake that is considered in the analysis must also be specified.The LBM limits the earthquakes that are considered in assessing the probability that specified ground motion levels are exceeded. In the past there has not been a direct consideration of the appropriate LBM value that should be used in a seismic hazard assessment. This study specifically looks at the selection of a LBM for use in seismic hazard analyses that are input to the evaluation/design of nuclear power plants (NPPs). Topics addressed in the evaluation of a LBM are earthquake experience data at heavy industrial facilities, engineering characteristics of ground motions associated with small-magnitude earthquakes, probabilistic seismic risk assessments (seismic PRAs), and seismic margin evaluations. The results of this study and the recommendations concerning a LBM for use in seismic hazard assessments are discussed.     

Stochastic prediction of maximum seismic response of light secondary systems

A stochastic model is presented for predicting the elastic response of light multi-degree-of-freedom secondary systems to strong motion earthquakes. Secondary systems may include light mechanical or electrical equipment, piping, or other light systems attached at one or several points to walls or floors of the supporting or primary structures. The critical functions of these secondary systems in nuclear power plants make the accurate prediction of their maximum responses important. The response of such secondary structures may be obtained by a direct time-history analysis, or more approximately, by the response spectrum method. The time-history solution is, of course, expensive; moreover, there is no single representative earthquake and thus a number of possible earthquake ground motions have to be considered. On the other hand, the response spectrum method applied to secondary systems can lead to unreliable results.Within the framework of the normal mode method, a decoupled stationary random vibration model is developed based on the assumption of Gaussian response process and Poisson barrier crossings. The accuracy of the proposed model is verified by comparing the calculated responses, at the 10 and 50% probability of exceedance level, with the second highest and average of the time-history responses from eight normalized accelerograms. The influence of decoupling, i.e. ignoring the dynamic interaction between the primary and the secondary systems, on the response is examined.The influence of nonstationarity is also evaluated. It is observed that nonstationarity is unimportant for earthquakes of relatively long duration, and that for a given damping most of the error can be accounted for by a simple scaling. It is also shown that one aspect of the proposed method constitutes the basis for some of the approximations in the response spectrum method; however, the proposed method yields results that are consistently more reliable than the response spectrum method. Moreover, results obtained with the proposed method represent maximum response statistics from an ensemble of earthquakes rather than a single earthquake.     

Consideration of uncertainties in seismic analysis of non-classically damped coupled systems

This paper proposes a method to evaluate the design response of a non-classically damped coupled primary-secondary system by statistically incorporating the effects of uncertainties in modal properties of its constituent uncoupled systems. Within the framework for the coupled system seismic analysis, the uncertainties can be accounted for by modeling the uncoupled modal properties of primary and secondary systems as random variables. Gupta and Choi (2005) proposed the Square-Root-of-Mean-of-Squares (SRMS) method which employs a limited Monte Carlo simulation to evaluate the design response of the secondary system statistically. The SRMS approach was illustrated to work well with representative single degree of freedom (SDOF) primary-SDOF secondary systems. In this paper, we study the applicability of SRMS methodology to MDOF primary-MDOF secondary systems. In such systems, two or more modes are likely to have closely spaced frequencies. The individual probability density functions of the closely spaced frequencies overlap with each other. Simulation of such closely spaced frequencies as independent random can give incorrect set of frequencies in the sense that the frequencies do not remain as ordered sets. Rejection of these incorrect sets does not resolve the problem as the simulated density functions no longer maintain the originally assumed distribution. The simulation of ordered sets of natural frequencies of an MDOF structure can be achieved by using a joint density function that considers the necessary constraints. The SRMS method for MDOF primary-MDOF secondary coupled systems is modified by incorporating a closed-form formulation for the joint density function of closely spaced frequencies. The modified SRMS approach is validated for MDOF secondary systems that are both singly as well as multiply connected to the MDOF primary system.