Complex modal properties of coupled moderately light equipment-structure systems

There are many practical problems in which the damping matrix cannot be represented in a classical form, thus leading to complex mode shapes and frequencies. A modal superposition method making use of the complex mode shapes is presented. In the method a complex mode shape is replaced by two real modal vectors. The modal equations are integrated in the same fashion as in the classically damped systems, except we now also use the modal velocities in addition to the usual modal displacements. It is shown that the modal superposition method for the nonclassically damped system gives response values almost identical to those given by the direct integration method. A method of constructing the damping matrix for a coupled primary and secondary system is also presented.     

The uncoupling criteria for subsystem seismic analysis

This paper discusses the uncoupling effects of a subsystem from the system based on frequency, mode shape and response variations. The two-mass system is first used to study the problem, and a closed form solution of the frequency variation is derived for the two resonant masses with different mass ratios. Since the coupled and uncoupled analyses have a different number of modes, proper selection of modes for frequency variation check is also discussed. The resonance effect of a coupled analysis is shown in the mode shapes. Thus, the closed form solution of mode shapes for two resonant masses is also derived as a function of mass ratio. Since the response variation is a function of input, the response variation of a two-mass system subjected to white noise input is discussed.Since multiple degree of freedom systems are used in the majority of cases, the results of the two-mass system are extended to the multiple degree of freedom systems by the concept of normal modes. Each normal mode can be represented by a single degree of freedom system with equivalent modal mass and equivalent modal spring. Different equations were used in the past to define the equivalent modal masses depending on the objective of the analysis. The method of defining the equivalent modal mass which takes into account the location of a subsystem, is recommended. Once the equivalent two-mass system of the multiple degree of freedom systems and subsystems is derived, the frequency, mode shape, and response variations of the multiple degree of freedom system and subsystem can be assessed.The above coupling/uncoupling analysis is applied to two different situations. The first one is the building-equipment interaction usually with small mass ratios. The second one is the equipment-equipment interaction where the mass ratio can be large. Here, the equipment includes piping systems, pressure vessels, pumps, etc. The uncoupling analysis of the first case is required because the equipment information is not available during the building analysis. The uncoupling analysis of the second case is required due to practical need to reduce the size of the model. The recommendations of the uncoupling analysis of both cases are presented.     

A perturbation analysis of the eigenproperties of equipment-structure systems

In the calculations involving the dynamic response of equipment, it often is of interest to include the effect of the dynamic interaction between the equipment and its supporting structure. This can be done by calculating the eigenproperties of the combined equipment-structure systems, through either a mode synthesis approach or a perturbation approach. Herein, the details of a systematic perturbation expansion scheme are given for the calculation of the combined modal properties. Ready-to-use closed form expressions are provided for calculating the frequencies, mode shapes and participation factors, both for a detuned as well as a tuned equipment. These expressions can be used for equipment which are not heavier than 1/10 the mass of the supporting floor. However, if only the combined frequencies are desired the expressions can also be utilized for equipment as heavy as one-half the mass of the supporting floor without much error.     

Seismic response of a structure subjected to rotational base excitation

A modal superposition method which can perform the seismic analysis of a structure subjected to translational and rotational base excitation is presented. Discussed are two different approaches to derive the equations of motion. In the first approach, the reference axes are fixed in space. While in the second approach, they are rigidly fixed at the base of the structure. For rotational base excitation, it is shown that the application of second approach results in equation of motion with asymmetric, time-dependent coefficient matrix due to presence of the Coriolis acceleration term.Analytic integration is used to integrate the modal equations of motion derived by the first approach. Most of the mode shapes have to be included in the time history analysis.The modal superposition method is applied to the seismic analysis of a building subjected to translational and rotational excitation. The displacement results and the computer cost of this analysis are compared with those of using the direct integration method. The computer cost associated with the modal superposition method is lower than that associated with the direct integration method.     

PWR internals vibrational mode shapes calculation and tests

In the case of Pressurized Water Reactors (P.W.R.) the core is contained in a cylindrical core barrel, surrounded by a thermal shield, with its top rim at the vessel flange. The coolant flow along those internals is highly turbulent and induces some structure vibrations.

The barrel and thermal shield motion change more or less the neutron transmission through the water annulus : consequently, the resonant frequencies of structural vibrations may induce resonances on the power spectral density of ion chamber current. The relative amplitude of the neutron noise and mechanical resonances depends on the mode shapes. A good knowledge of these modal shapes is necessary to make a good interpretation of neutron noise result.

Obtained from a french computer system called AQUAMODE-TRISTANA, using the modal coupling of substructures, taking into account the liquid effects by finite elements and using experimentally deduced turbulent pressure forcing functions, the resonant frequencies, modes shapes and amplitudes of P.W.R. internals (900 MW, three loops) are presented in the paper.

These results are in good agreement with experimental results obtained on the SAFRAN Loop which consists of a reduced scale model of three loops P.W.R. primary circuit and reactor internals and on the FESSENHEIM reactor during cold flow tests.     

Seismic analysis of a reactor coolant pump by the response spectrum method

Current nuclear steam supply systems (NSSS) are designed to remove the heat of fission by circulating coolant in closed loops from the reactor. For water reactors, this prime function is designated to the reactor coolant pump (RCP). The Westinghouse Type 93A RCP is analyzed for seismic response. Briefly described, this RCP is a vertical, single-stage, centrifugal pump designed to move 90 000 gpm (568 m³/sec) of water and driven by a 6000 hp motor for use in the PWR primary system. The RCP assembly is generally axisymmetric and is modeled using three-dimensional finite elements of the types normally found in general-purpose computer programs such as ANSYS or NASTRAN. The structural frame and the rotating shaft are the principal branches of the model. Each consists of a series of pipe elements complemented by mass elements. Orthogonal sets of linear spring elements connect the branches at the bearings and possibly at each labyrinth. Fluid elements are added to include the interaction between the shaft and the pump case through the intervening water mass. Beam elements are used to account for unsymmetry of the motor stand. To complete the model, stiffness matrix elements representing the support structure and the neighboring loop piping are attached. It is impractical to idealize faithfully each geometric irregularity. Several adjacent sections are combined into one suitable element with total stiffness and equivalence. The number of elements in the model is thus minimized. Shear deflection of the pipe elements is considered; mass and mass inertia are lumped at nodal points, as needed to compensate for the actual material distribution. The RCP model contains 82 nodes, 155 elements and 140 master dynamic degrees of freedom. A modal frequency analysis is first run to identify the mode shapes.The seismic analysis is performed by the response spectrum method in ANSYS, with seismic velocity as the input excitation parameter. The model is excited by a set of three orthogonal spectra. For each load excitation, the modal displacements, forces and moments are computed at each node. A post-run subroutine calculates the absolute sum of nodal response quantities at each mode for one horizontal and the vertical seismic excitations. The resultant modal values are then combined using the square root of the sum of the squares (RSS) to record the final values: SSE X-Y and SSE Y-Z. Nodal stresses are computed; absolute displacements are reviewed for selected nodes along the model branches. The relative displacements at bearings and labyrinths are determined. Finally, the accelerations of nodes previously chosen are found.This paper assesses the effects of a given seismic excitation on the overall structural integrity of an RCP. The in-depth analysis has found the RCP adequate to withstand the imposed seismic loading. All component stresses are within the applicable faulted criteria and the relative movements between closely mated parts fall inside their nominal clearance limits.     

Seismic response of multiply connected MDOF primary and MDOF secondary systems

New algorithms are presented to evaluate the mode shapes and frequencies of a coupled system, given the mode shapes and frequencies of the uncoupled primary and secondary systems. These coupled mode shapes can be used to obtain the dynamic response of the total system given the input to the primary system. This information can also be used to develop instructure response spectra (IRS) at the connecting DOF, along with the correlation between the motions at the connecting DOF.     

Analysis of a cylindrical shell vibrating in a cylindrical fluid region

Analytical and experimental methods are presented for evaluating the vibration characteristics of cylindrical shells such as the thermal liner of the Fast Flux Test Facility (FFTF) reactor vessel. The NASTRAN computer program is used to calculate the natural frequencies, mode shapes, and response to a harmonic loading of a thin, circular cylindrical shell situated inside a fluid-filled rigid circular cylinder. Solutions in a vacuum are verified with an exact solution method and the SAP IV computer code. Comparisons between analysis and experiment are made, and the accuracy and utility of the fluid-solid interaction package of NASTRAN is assessed.     

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.     

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.     

核电站主泵机组地震响应谱分析及应力评定

用三维实体建模软件Inventor建立某核电站主泵的三维实体模型。对模型进行简化,灵活运用ANSYS的单元属性和接触功能,建立有限元动力学模型。通过模态分析,得出前13阶固有频率。在此基础上,用SRSS振型组合法分析多地震谱、多角度下核主泵的地震谱响应,得到了相应的应力和位移响应。对主泵进行静力学分析,将地震动应力与静应力相叠加,分析不同工况下主泵机组的应力值。按ASME规范进行校核,结果表明：应力值满足标准要求。     

Use of power spectral density in deriving response spectral shapes for aseismic design

Mean and mean + σ response spectral shapes based on accelerograms normalized to their respective peak ground acceleration (PGA) values have been widely used in the aseismic design of critical structures such as nuclear power plants. The computation of response spectral shapes based on accelerograms normalized to their respective power spectral densities (PSDs) has been proposed. A comparison of response spectral shapes using the two alternative normalization parameters (PGA and PSD) has been presented to demonstrate that the ordinates of the PSD-based mean + σ spectral shapes, which are better representatives of earthquake vibrations, are considerably lower than those of the PGA-based spectral shapes. The mean spectral shapes in the two cases are very close to each other, suggesting that a large part of the scatter in the PGA-based shapes is attributable to the procedure adopted for normalization of the accelerogram. The PSD-based spectral shapes can play an effective role in determining the margins available in the aseismic design of structures using the PGA-based mean + σ spectral shapes. Smooth response spectral shapes based on normalization with respect to the PSD are given for rock and soil sites at the end of the paper.     

Consideration of uncertainties in seismic analysis of coupled building piping systems

In this paper, we present an analytical study for incorporating the effect of uncertainties in modal properties of uncoupled primary and secondary systems in the seismic analysis of non-classically damped coupled systems such as building piping by response spectrum method. Monte Carlo simulation is used to illustrate that the secondary system design response when defined at a non-exceedence probability of 0.84 over the individual responses obtained from multiple response spectrum analyses by considering uncertainties in modal parameters is excessively higher than the design response specified at the same non-exceedence probability over the responses obtained from multiple time history analyses. This is so because the earthquake input in a response spectrum method is characterized by a design spectrum which by itself is specified at a non-exceedence probability of 0.84 over the multiple time histories with normalized peak ground acceleration. Accurate evaluation of design response at a non-exceedence probability of 0.84 in the response spectrum method requires that the individual modal responses be defined at appropriate probability levels that may be different than the conventionally used non-exceedence probability value of 0.84. The required probability values are evaluated by using first order reliability method. It is shown that the modal responses, when defined at a non-exceedence probability of 0.84, would give relatively accurate values of design response only if the individual modes are perfectly correlated or a single mode contributes to the particular response quantity of interest. For all other cases, the design response would be excessively high. The accurate probability values needed to specify each modal response evaluated using the first order reliability method cannot be incorporated directly in a response spectrum analysis due to computational inefficiency. Two simplified methods, based on total probability theorem, are developed in this paper to overcome this limitation. It is shown that these methods give design response values that are very close to the true values obtained from multiple time history analyses.     

Simulation of dynamic environments for design verification

Surveys on verification techniques are updated in this report to include recent applications in nuclear facilities that use transient methods for dynamic excitation. Also covered are brief discussions of the theoretical considerations underlying this testing approach, as well as requirements for correlation of experimentally and analytically derived data. Within the context of the experimental approach, the application of the rapidly changing technology of digital signal capture and data processing portends shorter test times, lowered costs, and more comprehensive measurements and environmental simulations.The experimental data encompass the use of structural transfer functions. This type of data may be used to extract mode shapes, modal frequencies, and damping, and can also be used directly to calculate structural system response motions to a variety of dynamic forcing functions.Force pulse generators provide large input forces over short time periods for in-place testing of very large structures and massive equipments. The pulse generators are applied in two different ways. In one they are used to excite structures for measurement of transfer functions; in another, now being developed, multiple pulse generators are attached to large structures to create a response that will closely match that predicted for the structure when subjected to an earthquake or other dynamic loads.The discussions culminate in a projection of the design and development work that will be required to meet future needs in structural response of nuclear reactors to dynamic loads. It is envisioned that the technical community will be required to focus attention on nonlinear behavior with special emphasis on how structures can be tested to yield their structural response characteristics.     

Analysis of piping with hysteretic supports using response spectra

This paper introduces a response spectra-based method for analyzing piping with hysteretic nonlinear supports. The method is developed to be as simple and versatile as possible, yet accurate enough to model the essential nonlinear behavior of the supports. The required data is the hysteresis loops of the supports, the linear properties of the piping, and the linear acceleration response spectra. The supports are modeled by equivalent linear stiffness and damping, and the combined piping/support system is analyzed using complex modal properties that account for high-damping effects. The final peak response is obtained by a mode combination rule which is a new generalization of Complete Quadrature Combination (CQC) that accounts for nonlinear properties and complex modes. Sensitivities that indicate the degree of nonlinear behavior and support interaction are also determined. The method is used to analyze two three-dimensional piping systems with multiple nonlinear supports, which have been tested on a shaking table. Comparisons between experimental and analytical results show good agreement.     

Vibration analysis of a dummy fuel rod continuously supported by spacer grids

The fuel rods in the pressurized water reactor are continuously supported by a spring system called a spacer grid (SG), which is one of the main structural components for the fuel rod cluster (fuel assembly). The fuel rods have a vibration behavior within the reactor due to coolant flow. Since the vibration, which is called flow-induced vibration, can wear away the surface of the fuel rod, it is important to understand its vibration characteristics. In this paper, a modal testing and a finite element (FE) analysis using ABAQUS on a dummy fuel rod continuously supported by Optimized H Type (OHT) and New Doublet (ND) spacer grids are performed to obtain the vibration characteristics such as natural frequencies and mode shapes and to verify the FE model used. The results from the test and the FE analysis are compared according to modal assurance criteria values. The natural frequency differences between the two methods as well as the mode comparison results for the rod with the OHT SG are better than those with the ND SG. That is, in the case of the ND grid model using beam-spring elements, there was a large discrepancy between the two methods. Thus, we tried to modify the FE model for the ND SG considering the contact phenomena between the fuel rod and the SG. The results of the new model showed a good agreement with the experiment compared with those of a beam-spring model.     

Forced vibration tests and analyses of nuclear power plants

Methods for conservatively predicting the response of a constructed nuclear power plant to earthquake excitations are presented. This approach is based on experimental testing of the reactor plant and using test results to develop a mathematical model of the system. First, steady state forced vibration tests are conducted using structural vibrators attached to the reactor structure to determine dynamic response characteristics. Second, modal analysis applied on a digital computer is used to create a linear multiple-degree-of-freedom model that has dynamic response characteristics nearly the same as the physical system for the experimental inputs. Finally, the input force levels are extrapolated from the levels of the inertial vibrators to earthquake levels and the response of the model is calculated for strong-motion earthquakes.Tests have been conducted on three nuclear power plants: the experimental gas-cooled reactor (EGCR) at Oak Ridge, Tennessee; the Carolinas-Virginia tube reactor (CVTR) at Parr, South Carolina; and the San Onofre Nuclear Generating Station (SONGS), San Onofre, California. Analyses in varying detail have been performed; the most extensive work has been done at San Onofre. This article summarizes test results, dynamic models, and the results of seismic response calculations for each plant.     

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.     

Three-dimensional dynamic response modeling of floating nuclear plants using finite element methods

A modelling technique which can be used to obtain the dynamic response of a floating nuclear plant (FNP) moored in an artificial basin is presented. Hydrodynamic effects of the seawater in the basin have a significant impact on the response of the FNP and must be included. A three-dimensional model of the platform and mooring system (using beam elements) is used, with the hydrodynamic effects represented by added mass and damping. For an essentially square plant in close proximity to the site structures, the three-dimensional nature of the basin must be considered in evaluating the added mass and damping. However, direct solutions for hydrodynamic effects with complex basin geometry are not, as yet, available. A method for estimating these effects from planar finite element analysis is developed.First, added mass and damping values are obtained from plane-strain finite element models of vertical cross sections through the platform. Fluid finite elements are used to model the seawater. For added mass calculations, the planar models include the platform cross section, the basin profile and the seawater in the basin. For hydrodynamic damping calculations, the planar model includes the platform cross section, the seabed and seawater, infinite in horizontal extent. Added mass and damping values are obtained for each significant mode of platform response. Estimates of three-dimensional added mass and damping are then obtained through combinations of the planar values. The release of the planar contraints of seawater motion and the reflection of gravity waves back to the platform are considered. Effective damping values applicable, on an average, for the entire response time are calculated for each plant mode of response. Since added mass and damping are frequency dependent, the selection of values to be used for a specific loading condition is usually an iterative process.The accuracy of the planar finite element model in obtaining two-dimensional added mass and damping is shown through comparison with existing and documented results. In addition, a comparison is shown for open ocean added mass and damping with a three-dimensional solution using velocity potential functions. It is concluded that the overall technique results in a reasonable and conservative calculation of the dynamic response of the floating nuclear plant.