WRC Bulletin Research Articles

Two and Three-Dimensional Analysis Comparison of Nozzles due to Internal Pressure, Thermal Load and External Load

본 논문에서는 원통형 쉘에 부착된 노즐의 구조 건전성평가를 수행하고 그 결과를 비교하기 위해 2차원(2D)과 3차원(3D) 해석이 수행되었다. 현재 원자력 발전소에서 사용되는 3개의 노즐을 구조 건전성평가를 위해 선정하였고, 각각 노즐은 내부압력, 온도변화 및 외부하중을 받는다. 내부압력에 대한 2D 해석은 1.5이상의 계수 값을 이용하거나 응력집중 계수를 적용하여야 하고, 온도변화에 대한 2D와 3D 해석결과는 피복재의 유무와 상관없이 서로 거의 비슷하며, 외부하중에 대한 WRC Bulletin 297에 의한 해석결과는 3D 해석결과보다 더 보수적임을 확인할 수 있었다.In this paper, the two-dimensional(2D) and three-dimensional(3D) analyses have been performed in order to evaluate the structural integrities and compare 2D and 3D results for nozzles attached to cylindrical shells. Three nozzles, which are currently used in the nuclear power plant, are chosen to evaluate the structural integrities, and each nozzle is subjected to internal pressure, temperature variation and external loads. It is found that the 2D analysis for internal pressure should be performed with a factor of more than 1.5 or a stress concentration factor; 2D and 3D analysis results for temperature variation are almost similar to each other regardless of cladding; and the analysis results for external loads by WRC Bulletin 297 are more conservative than the 3D analysis results.

Simplified Stress Linearization Method, Maintaining Accuracy

ASME PVP Code stress linearization is needed for assessment of primary and primary-plus-secondary stresses. The linearization process is not precisely defined by the Code; as a result, it may be interpreted differently by analysts. The most comprehensive research on stress linearization is documented in the work of Hechmer and Hollinger [1998, “3D Stress Criteria Guidelines for Application,” WRC Bulletin 429.] Recently, nonmandatory recommendations on stress linearization have been provided in the Annex [Annex 5.A of Section VIII, Division 2, ASME PVP Code, 2010 ed., “Linearization of Stress Results for Stress Classification.”] In the work of Kalnins [2008, “Stress Classification Lines Straight Through Singularities” Proceedings of PVP2008-PVT, Paper No. PVP2008-61746] some linearization questions are discussed in two examples; the first is a plane-strain problem and the second is an axisymmetric analysis of primary-plus secondary stress at a cylindrical-shell/flat-head juncture. The paper concludes that for the second example, the linearized stresses produced by Abaqus [Abaqus Finite Element Program, Version 6.10-1, 2011, Simulia Inc.] diverge, therefore, these linearized stresses should not be used for stress evaluation. This paper revisits the axisymmetric analysis discussed by Kalnins and attempts to show that the linearization difficulties can be avoided. The paper explains the reason for the divergence; specifically, for axisymmetric models Abaqus inconsistently treats stress components, two stress components are calculated from assumed formulas and all other components are linearized. It is shown that when the axisymmetric structure from Kalnins [2008, “Stress Classification Lines Straight Through Singularities” Proceedings of PVP2008-PVT, Paper No. PVP2008-61746] is modeled with 3D elements, the linearization results are convergent. Furthermore, it is demonstrated that both axisymmetric and 3D modeling, produce the same and correct stress Tresca stress, if the stress is evaluated from all stress components being linearized. The stress evaluation, as discussed by Kalnins, is a primary-plus-secondary-stresses evaluation, for which the limit analysis described by Kalnins [2001, “Guidelines for Sizing of Vessels by Limit Analysis,” WRC Bulletin 464.] cannot be used. This paper shows how the original primary-plus-secondary-stresses problem can be converted into an equivalent primary-stress problem, for which limit analysis can be used; it is further shown how the limit analysis had been used for verification of the linearization results.

Considerations in the Design and Analysis of an ASME Section VIII, Div. 2 Reactor Support Skirt

This paper describes the considerations employed in the finite element analysis of a relatively “short” support skirt on a hydrocarbon reactor vessel. The analysis is accomplished in accordance with ASME B&PV Code, Section VIII, Division 2 alternate rules in conjunction with the guidelines outlined in WRC Bulletin 429. This provides a sound basis for the classification of the calculated stress intensities. The support skirt is capable of sustaining the deadweight load in addition to resisting the effects of thermal displacements, wind loadings, overturning moments from external piping loads on the attached hydrocarbon reactor vessel, and friction between the skirt base plate and concrete foundation. The displacement and thermal boundary conditions are well defined and discussed in detail. The effects of multiple scenarios for the displacement boundary conditions are examined. The skirt design also employs a hot-box arrangement whereby the primary mode of heat transfer is by radiation. A discussion of the two-part analysis is included and details the interaction between the heat transfer analysis and the subsequent structural analysis. The heat transfer finite element analysis is utilized to determine the temperatures throughout the bottom of the vessel shell and head, as well as the integrally attached support skirt. Of prime importance during the analysis is the axial thermal gradient present in the skirt from the base plate up to and slightly beyond the skirt-to-shell junction. While the geometry of the subject vessel and skirt is best described as axisymmetric, the imposed loadings are a mixture of axisymmetric and non-axisymmetric. This combination lends itself to the judicious selection and utilization of the harmonic finite element and properly chosen Fourier series representation of the applied loads. Comparison of the thermally induced axial stress gradient results from the FEA to those obtained by the closed form beam-on-elastic-foundation are also tendered and discussed. Finally, recommendations are included for the design and analysis of critical support skirts for large, heavy-wall vessels.

Flexibility Factors for Branch Pipe Connections Subjected to In-Plane and Out-of-Plane Moments

Stress intensity factor and flexibility factor are important analysis parameters for branch pipe connections subjected to in-plane and out-of-plane moments. The calculation of stress intensity factors for a large range of unreinforced fabricated tees (0.333⩽d∕D⩽1, 20⩽D∕T⩽250, d∕D⩽t∕T⩽3) was performed by Widera and Wei [WRC Bulletin 497 2004] and empirical formulas were provided based upon a parametric finite element analysis employing four-node shell elements. The purpose of this paper is to extend the previous effort by Widera and Wei and calculate the in-plane and out-of-plane flexibility factors for the same range of geometric parameters. Similarly, empirical formulas for the determination of these flexibility factors are proposed. The results show that the lengths of the branch and run pipes as well as the geometric parameters (d∕D, D∕T and t∕T) have an effect on the calculation of flexibility factors.

Theoretical Stress Analysis of Intersecting Cylindrical Shells Subjected to External Forces on Nozzle

The stress analysis based on thin shell theory is presented for a cylindrical shell with a normally intersecting nozzle subjected to three kinds of branch pipe forces, which are tension and two shear forces. The basic mechanical models for the three load cases are presented in order to obtain the solutions independent of the length of the main shell and branch pipe. The applicable range of the present solution is expanded up to d∕D⩽0.8 and λ=d∕DT⩽12 by means of the accurate cylindrical shell equations and continuity conditions and the improved numerical method. The theoretical results are verified by test and three-dimensional (3D) finite element method (FEM) results. The maximum stress for tension force case are in good agreement with WRC Bulletin No. 297 when λ is small. The comparison between the present results and WRC Bulletin No. 107 shows that the latter needs improvement.

A Thin Shell Theoretical Solution for Two Intersecting Cylindrical Shells Due to External Branch Pipe Moments

A theoretical solution is presented for cylindrical shells with normally intersecting nozzles subjected to three kinds of external branch pipe moments. The improved double trigonometric series solution is used for the particular solution of main shell subjected to distributed forces, and the modified Morley equation instead of the Donnell shallow shell equation is used for the homogeneous solution of the shell with cutout. The Goldenveizer equation instead of Timoshenko’s is used for the nozzle with a nonplanar end. The accurate continuity conditions at the intersection curve are adopted instead of approximate ones. The presented results are in good agreement with those obtained by tests and by 3D FEM and with WRC Bulletin 297 when d∕D is small. The theoretical solution can be applied to d∕D⩽0.8, λ=d∕DT⩽8, and d∕D⩽t∕T⩽2 successfully.

Theoretical stress analysis of intersecting cylindrical shells subjected to external loads transmitted through branch pipes

A thin shell theoretical solution of two normally intersecting cylindrical shells subjected to thrust-out force and three kinds of moments transmitted through branch pipes is presented in this paper. The solutions of modified Morley equation, which can be applicable up to ρ 0 = d/ D ⩽ 0.8 and λ = d/( DT) 1/2 ⩽ 8 and the order of accuracy is raised to O( T/ D), for the four loading cases are given. The accurate continuity conditions of generalized forces and displacements at the intersecting curve of two cylindrical shells for the four loading cases and the condition of the uniqueness of displacements are derived in this paper. The presented results are verified by experimental and numerical results successfully. They are in agreement with WRC Bulletin 297 when d/ D is small.

Analytical solution of two intersecting cylindrical shells subjected to transverse moment on nozzle

In this paper a theoretical solution for two normally intersecting cylindrical shells subjected to transverse moment on the branch pipe is presented, which based on thin shell theory. The accurate shell equations, boundary conditions and calculating methods are adopted so that the solution presented can be applicable up to d/ D⩽0.8 and λ= d/( DT) 1/2⩽8. The presented results are in very good agreement with experimental and numerical results for ORNL-1 Model. They are also in agreement with the results obtained by WRC Bulletin 297 when d/ D is small.

External loads on nozzles

Local loads are still approximated quite often by means of the so-called ‘shrink ring’ method first published by the MW Kellogg Company in their publication ‘Design of Piping Systems’. In this article the shrink ring method is compared with calculation methods from WRC Bulletin 107 and BS 5500, Appendix G. Both nozzles on spherical shells as well as nozzles on cylindrical shells are taken into consideration. The stresses in spherical vessels were estimated quite reasonably by means of the shrink ring method; however, the stresses in cylindrical vessels are severely underestimated. An improved shrink ring method for cylinders is proposed in Appendix 2 of this article.