Inclusion of Local Shell Behavior of Tubes Into a Two-Dimensional Beam Approximation of Deformation in Nuclear Fuel Channels

A method has been developed to incorporate the local three-dimensional shell behavior of two concentric tubes in the two-dimensional beam modeling of the problem. The two dimensional modeling of fuel channels in CANDU pressurized heavy water nuclear reactors is used in lieu of a more accurate three dimensional finite element approach in order to reduce the on-line simulation time which greatly affects the SLAR (Spacer Location And Repositioning) maintenance operation cost during outage. However, effort must be made to include the three-dimensional shell behavior of these channels into the two-dimensional modeling. In recent studies a nonlinear force-dependent model for contact stiffness between the calandria tube and pressure tube has been developed. However, local deformation of calandria the tube at spacer locations due to in-reactor creep leads to settling of the spacer into the calandria tube that consequently reduces the gap between the two tubes. In this work, the effect of local deformation (elastic and creep) of calandria tubes on modeling of contact at spacer locations is assessed using a three dimensional finite element code. The result is incorporated into a two-dimensional beam model of the problem as a reduction in size of the spacers that separate the two tubes. It is shown that the proposed method increases the accuracy of prediction of contact time and the spacer. In general, the method described in this paper suggests a way to incorporate local shell deformation into beam models of slender shell structure.

The Meshless Dynamic Relaxation Techniques for Simulating Atomic Structures of Materials

Traditionally, Molecular Dynamics combined with pair potential functions or the Embedded Atom Method (EAM) is applied to simulate the motion of atoms. When a defect is generated in the crystalline lattice, the equilibrium of atoms around it is destroyed. The atoms move to find a new place where the potential energy in the system is minimum, which could result in a change of the local atomic structure. The present paper introduces new Dynamic Relaxation algorithm, which is based on explicit Finite Element Analysis, and pair or EAM potential function, to find equilibrium positions of the block of atoms containing different structural defects. The internal force and stiffness at the atoms (nodes) are obtained by the first and second derivatives of the potential energy functions. The convergence criterion is based on the Euclidean norm of internal force being close to zero when the potential energy is minimum. The damping ratio affects the solution path so that different damping ratios could lead to different minimum potential energy and equilibrium shapes. The numerical responses and results by applying free boundary conditions and certain periodic boundary conditions are presented. The choice of scaled mass of atoms, proper time step and damping appropriate for the efficient and stable simulation is studied.

Application of a Sixth Order Generalized Stress Function for Determining Limit Loads for Plates with Triangular Penetration Patterns

The capability to obtain limit load solutions of plates with triangular penetration patterns using fourth order functions to represent the collapse surface has been presented in previous papers. These papers describe how equivalent solid plate elastic-perfectly plastic finite element capabilities are generated and demonstrate how such capabilities can be used to great advantage in the analysis of tubesheets in large heat exchanger applications. However, these papers have pointed out that although the fourth order functions can produce sufficient accuracy for many practical applications, there are situations where improvements in the accuracy of in-plane and transverse shear are desirable. This paper investigates the use of a sixth order function to represent the collapse surface for improved accuracy of the in-plane response. Explicit elastic-perfectly plastic finite element solutions are obtained for unit cells representing an infinite array of circular penetrations arranged in an equilateral triangular array. These cells are used to create a numerical representation of the complete collapse surfaces for a number of ligament efficiencies (h/P where h is the minimum ligament width and P is the distance between hole centers). Each collapse surface is then fit to a sixth order function that satisfies the periodicity of the hole pattern. Sixth-order collapse functions were developed for h/P values between .05 and .50. Accuracy of the sixth order and the fourth order functions are compared. It was found that the sixth order function is indeed more accurate, reducing the error from 12.2% for the fourth order function to less than 3% for the sixth order function.

Computation of Ratchet Limits for Structures Subjected to Cyclic Loading and Temperature

The paper discusses methods of evaluating the ratchet limit for an elastic/plastic structure subjected to cyclic thermal and mechanical loading. A recently developed minimization theorems by Ponter and Chen [2] provides a generalization of the shakedown limit theorems for histories of load in excess of shakedown. This allows the development of programming methods that locate the ratchet boundary in excess of shakedown. Examples of applications are provided including the performance of a cracked body subjected to cyclic thermal loading. Finally, the theory is used to discuss Kalnins’ [4] proposal that short cut finite element solutions may be used to assess whether a particular loading history lies within a ratchet limit.

Numerical Study of Mass Dispersion in Laminar and Turbulent Flows Through a Pipeline

A computer code for solving the equations of mass diffusion has been developed and applied to study the molecular-level mixing between two fluids inside a pipe. First, one fluid occupies the entire volume within the pipe, and then a second miscible fluid is forced into the pipe, developing a mixing process through the interface between the fluids. This phenomenon occurs as the combination of molecular diffusion, variation of velocity over the cross-section and turbulence. The code developed for this study is based on the finite element method for domain discretization and standard finite difference schemes for temporal discretization. Comparison with experimental data shows that the code is able to reproduce the physical trends and gives good predictions for engineering applications. A grid independence analysis is presented for all computations.

Piping System Structural Integrity Simulation for Post LOCA Water Hammer Loads

The Canadian Nuclear Safety Commission (CNSC) required as part of the operating license for Ontario Power Generation’s Darlington Nuclear Generating Station, that the structural integrity of the piping following a loss of coolant accident (LOCA) be demonstrated. This is necessary to ensure that no subsequent pressure boundary failures will impede the ability to maintain fuel cooling. The injection of cold emergency coolant following a LOCA creates the potential for the occurrence of condensation-induced water hammers (CIWH) in the primary heat transport (PHT) system piping. Classical linear elastic piping analysis using the class 1 NB-3656 rules of the ASME Boiler & Pressure Vessel Code failed to demonstrate the adequacy of the piping and/or its supports that were designed using the linear elastic rules of subsection NF for nine of the twelve piping models that comprise the PHT system. A decision was made to undertake a state-of-the-art non-linear explicit analysis in order to qualify the piping. Strain rather than stress limits would be applied similar to those being developed by ASME for nuclear packaging undergoing accidental impact during transportation. In order to address the feasibility of this approach, a non-linear analysis was performed on a portion of one of the piping systems. The piping was modeled as shells and again as beam elements with and without detailed modeling of the supports. After these initial simulations, it was determined that the piping could be modeled with simplified beam elements, however, the supports would require a more detailed modeling in order to determine the extent of support damage and the effect the supports have on the integrity of the piping system itself. This paper addresses the non-linear modeling of the piping models and discusses the modeling details, assumptions and analysis results. This approach is shown to be a useful alternative for predicting the extent of structural damage that can be expected by a Level D event such as a condensation induced water hammer following a loss of coolant accident.

Extended GDQ and Related Discrete Element Analysis Methods for Transient Analyses of Continuum Mechanics Problems

The extended GDQ proposed by the author is used to develop solution algorithms for solving the discrete transient equation system of continuum mechanics problems. It is a direct integration approach. Two integration methods are developed. They are time-element by time-element method and stages by stages method. These two time integration algorithms can be used to solve generic discrete transient equation system of an originally discrete system or a discrete system resulting from the discretization of a transient system of continuum mechanics problems by using a certain discretization technique such as the DQEM, FEM, FDM,…, etc.

Some Achievements of the European Project LISA for FEM Based Limit and Shakedown Analysis

The load-carrying capacity or the safety against plastic limit states are central questions in the design of pressure equipment. The new European standard prEN 13445-3 reflects the fact that plastic limit states can be calculated simply by perfectly plastic limit and shakedown analyses. These methods can be derived from static and kinematic theorems for lower and upper bound analysis, respectively. Both may be formulated as optimization problems for finite element discretizations of structures. The problems of large-scale analysis have been solved in an European research project. The methods could be made more realistic by extension to other material models such as a two-surface plasticity model of kinematic hardening. Limit and shakedown analyses are briefly demonstrated with illustrative examples, which are chosen to show some limitations of the 3Sm criterion and the effect of kinematic hardening.

Equivalent Solid Based Fatigue Analysis of Perforated Plates with Triangular Perforation Pattern

The current fatigue assessment method of the ASME B&PV Code uses an elastic stress multiplier to determine the total stress in perforated plates with triangular pattern. This method contains several simplifications and shortcomings. The present paper shows that the perforation pattern symmetry imposes constraints on the total stress multipliers that can be used to simplify the elastic analysis, particularly of three-dimensional (non-axisymmetric) plates. For thin ligament conditions, the very conservative Code analysis can be replaced by a more accurate stress multiplier approach for which the stress concentration around the hole is derived from an elastic Finite Element analysis. This can result in a significant reduction of the calculated fatigue usage factor. Finally, the issue of strain concentration is addressed. When the total stress range at a location around the ligament exceeds 2Sy, the elastically predicted strain range is potentially unconservative. The elastic-plastic strain at this location can be estimated from the elastic result using simplified methods. The corresponding predictions are compared to elastic-plastic analysis results.

Subcycled Hourglass Control for Explicit Time Integration of Dynamic Relaxation Equations

Explicit methods, such as the central difference operator, rely on the economical evaluation of internal forces at each time step of a transient dynamic problem. One-point quadrature applied to the spatial discretization provides the greatest efficiency, but hourglass control is required to eliminate spurious zero energy modes. Computationally practical hourglass control methods involve considerable approximation in evaluating the internal force. Thus, a small additional approximation due to an alternative temporal integration of the hourglass force may not seriously affect the accuracy of the analysis. In particular, the possibility of evaluating the hourglass terms on a larger time interval than the usual stable time step could provide significant efficiencies. The proposed approach of subcycling the hourglass terms is examined in detail with respect to stability and accuracy. Implementation into an explicit finite element program is demonstrated on a three-dimensional example that involves several hourglass modes, and the new method proves to be beneficial for noninertial problems where artificial damping is used.