Corresponding author: Artem V. Sobolev (SobolevArtem82@gmail.com)

Academic editor: Yury Kazansky

Results of calculation seismic resistance analysis of light equipment of nuclear power plants performed on the example of a ventilation unit using two most common analytical techniques - linear spectral analysis and direct dynamic methods - are discussed.

The basic concepts, assumptions and limitations of the linear spectral method are described. Examples are given of specific calculation cases when the method in question is not applicable in the generally accepted formulation. In particular, the phase difference and, possibly, accelerations (displacements) must be taken into consideration in the calculations of extended spatial structures for mutually remote boundary conditions. Another example are the reservoirs not completely filled with liquids. In such case waves may be formed in the liquid and taking them into account is not possible in the linear spectral method.

Specific features are examined of application of the dynamic analysis method including the input data, approaches and methodologies required for synthesizing the calculated accelerograms. A sequence of operations performed during synthesizing calculated accelerograms is provided, materials are provided containing the description of the mathematical apparatus applied for deriving the final mathematical relations for calculating response spectra and the calculation relations as such are given. The concept of the damping coefficient is explained, its influence on the calculated results and the approaches to its determination are demonstrated. Options with complete absence of damping and with absolute damping are discussed.

A real ventilation set applied in active ventilation systems of nuclear power plants was accepted as the test model. Results calculated for the detailed finite-element model of the ventilation unit using the Zenith-95 software package are presented. These results include the distribution of the calculated reduced stresses. Analysis of the results obtained using the two methods demonstrated overestimation of calculated results by the linear spectral method as compared to those obtained by the dynamic analysis method, which means that the former method underestimates the equipment’s resistance to seismic effects. In addition, the dynamic method shows additional areas in the ventilation unit where significant reduced stresses are found while the linear spectral method ignores these areas.

Modern requirements to the design of equipment and pipelines of NPPs and other objects where nuclear energy is used include the requirements on seismic resistance (

Comparison of results of seismic resistance analysis performed using two most widely applied methods is of immediate interest. Speaking about heavy equipment, it was demonstrated in (

At present the following two calculation methods are the most widely spread: method of direct dynamic analysis and linear spectral method. There exist other, less widely spread, methods, for instance, the method described in (

The first among the above indicated methods presupposes solution of the differential equation describing displacements of the point by the numerical integration methods (

where

In current practice the above set of equations is solved using the finite-element method searching spatial distribution of displacements and other desirable functionals. Information on the displacements in the course of seismic impact (X(t) is the time dependence of displacements of supports) is used as the input data in the solution of Eq. (1). However, accelerograms of accelerations, i.e. the dependence of acceleration on the supports on time, are the most often used. Here, several important assumptions are made: each support behaves as the single rigid body (

Damping (dissipation of energy) is preset in the damping matrix and determines to a significant extent the results of calculation analysis. Acting regulations recommend setting the value of the damping coefficient to be equal to 2% for all elements and structural nodes. It is important that the results of calculation analysis are determined to a significant extent by the damping coefficient. If the coefficient is preset to be equal to zero, then all energy deposited by the seismic impact is conserved in the analyzed structure (i.e. it is spent on the displacements of its elements). Along with dissipation of energy damping determines the shift of eigenfrequences, which is of significance.

Calculation of the damping coefficient is a separate complex task, solution of which is, in particular, discussed in (

Direct dynamic analysis method is realized in the majority of software complexes. This instrument of analysis of seismic resistance allows taking into account all types of nonlinearities of the structure under analysis. All kinds of contacts (

Linear spectral method is currently the most widely used. It is widely applied both in Russia and abroad and represents, in general terms, the symbiosis of dynamic and quasi-dynamic analyses. Affinity to the quasi-dynamic method resides in the fact of calculation of seismic inertial loads _{i}_{i}_{i}_{i}

Seismic impact is determined for each form of seismic vibrations from the following relation:

where _{ij}_{ij}_{ij}_{i}_{i}_{i}

Formula (2) is derived from the equation for vibrations of arbitrary multi-mass system. The main assumption in its derivation is the small effect of damping on the relation between the forms of vibrations. This condition is satisfied in the examination of structures when for all elements the inelastic resistance coefficient is below 0.3.

Spectral method is accepted at present as the main method in regulatory documents acting in construction engineering (

Despite the abundant assumptions inherent to the linear spectral method, this methodology demonstrates satisfactory agreement with observed results. However, within the framework of this method there exist different approaches to the determination of the final seismic load knowing the loads for different forms of vibrations (

Significantly lower expenditures of computational resources for performing calculations are the main advantage of the linear spectral method. This method was often applied for performing manual calculations without the use of computers (

One of the problems emerging in the use of the dynamic method for seismic resistance analysis of equipment and pipelines is the lack of accelerograms and information about displacements in the process of seismic impact. As a rule, calculations with determination of seismic response spectra for different elevations of installation of equipment are performed within the framework of development of project of seismically resistant building.

Dynamics of seismic impact on the ground surface of the base can be selected on the basis of comparison of geological conditions and estimation of the distance to the nearest epicenter of the earthquake according to the data of seismic observations. However, such approach is absolutely not applicable to the presetting seismic impact on the equipment installed inside the building. Dynamics of construction structures must be taken into account here. Unfortunately, Russian national regulations in nuclear power engineering do not contain easily understandable recommendations on the synthesis of accelerograms for analysis of seismic resistance of equipment. The only guidance is provided in (

Two different approaches are used in construction for obtaining seismic impact accelerograms: synthesis on the basis of a package of close accelerograms registered during observations (

The following conditions must be satisfied in case when the package of accelerograms is applied: 1) representability of the accelerograms in the package, which amounts to the overlapping by the prevailing period of the whole range of variation of possible periods of seismic impact; 2) absence of serious distortions in the calculated accelerograms; 3) taking into account correlation between the calculated degree of intensity, magnitude and prevailing frequency of the impact. Synthesized accelerogram must produce after integration seismic response spectrum close to that preset in the design project.

Synthesis of accelerograms using generation of random process during the short interval of time found its application along with the synthesis of the calculated accelerogram using the data from the package of real accelerograms. Already at the dawn of elaboration of seismic resistance theory its founders in the name of Professors Mononobe and Suehiro performed modeling of seismic impact using section of the function

Modern methodologies (

Design of real radial ventilation unit applied in the special ventilation systems of NPPs was used for comparison of calculations performed by the spectral and dynamic methods. Calculations were performed by the finite-element method for the detailed model of the ventilation unit developed in the Zenith-95 software complex (

Model of the ventilator

The material of the ventilator model is stainless steel. Internal diameter of the round flange if equal to 500 mm. Upper flange is square-shaped with internal dimension equal to 350 mm. The motor in the model is represented as the absolutely rigid body with mass equal to 18 kg. The motor is connected to the fan propeller via the shaft with diameter equal to 24 mm. Internal pressure was not preset. The ventilator is rigidly fixed to the lower nodes of the rack frame. Coupling of the elements was achieved by the following two methods. In the first case the parts were coupled by way of “joining” the nodes, and in the second case it was achieved by the parts interacting via the “rigid” contact (analogue of welding) preset over the surfaces of the joint.

Spectra of interactions of the real building of the NPP in two mutually perpendicular directions and in one vertical direction were used for the calculation. Spectrum of seismic impact in the vertical direction is represented in Fig.

Designed spectrum of seismic impact. Vertical component.

Synthesized accelerogram of the seismic impact. Vertical component.

Maximum reduced stress in the ventilator in the analysis of seismic resistance by the method of direct dynamic analysis amounts to 24.2 MPa (Fig.

Amplitude values of reduced stresses calculated for the ventilator by the dynamic analysis method, MPa.

Time variation of calculated stresses in the critical node.

Amplitude values of reduced stresses of the ventilator calculated by linear spectral method, MPa.

It follows from the obtained results that spectral method using integral characteristics of the seismic impact produces overestimated calculated values of stresses, thus underestimating the real characteristics of seismic resistance of the equipment. In other words, when strength criteria are to be substantiated for the equipment for the seismic impact using the results of linear spectral calculation it will be guaranteed that these criteria will be satisfied (by wide margins) when dynamic analysis is applied. Despite the fairly similar picture of stresses, additional stressed areas are present in the ventilator model calculated using the dynamic method. Presence of these stressed areas indicates the advantage of the method in question consisting of the more detailed description of operation of the structure during seismic process.

If computer time resources required for performing calculations by the two methods are compared, then the CPU time needed for performing the dynamic calculation is by one – two orders of magnitude higher than that for the spectral calculation.

Thus, when detailed estimation of the behavior of the structure or its separate elements during seismic impact or in the course of analysis of especially important equipment (reactor vessel, steam generator, etc.) is needed, then the method of direct dynamic analysis should be applied. Application of linear spectral method underestimating seismic resistance due to the shifting of calculated functionals characterizing seismic resistance at its worst will be sufficient in case when seismic stability must be estimated as a whole without taking a close look at the specific features of dynamics of the impact from seismic process on the structure under analysis with economy of expenditures of computational and time resources.

✩ Russian text published: Izvestiya vuzov. Yadernaya Energetika (ISSN 0204-3327), 2018, n. 3, pp. 18–29.