On the Donnell's equations in the creep range

This paper is concerned with assessing the accuracy of Donnell's approximation when employed in the creep analysis of a class of circular cylindrical shells. Basic formulation of a general method describing the creep behaviour of two-dimensional cylindrical shells is first presented. The terms affected by Donnell's approximation are then pointed out. The solution of governing equations is obtained through coupling the ‘extended Newton's method’ and finite difference technique in an iterative procedure. A number of examples having geometries falling within the shallow shell definition, around the limit, and beyond the range of applicability, are solved using both theories. It has been noted that the parameter α, representing the shell geometry, has a pronounced effect on the accuracy of Donnell's simplification. As α increases the deviation between the two theories decreases. It is concluded that for the class of circular cylindrical shells considered herein Donnell's approximations yield accurate results for creep analysis, particularly for higher values of creep exponent n. Of course, employing Donnell's approximations results in simpler formulation and a reduction in computational time.     

A faster procedure for the calculation of the J(ξ, β) function

One of the biggest difficulties in obtaining an analytical expression for the J(ξ, β) function is its explicit dependence on the Doppler broadening function ψ(x,ξ). The objective of this paper is to present a method for the fast and accurate calculation for the J(ξ, β) function based on the recent advances in the calculation of the Doppler broadening function and on a systematic analysis of its integrand. The methodology proposed uses an analytical formulation for the calculation of ψ(x, ξ) and a representation in series for error functions with complex argument. The results were satisfactory from the accuracy and processing time standpoint and are an option to other calculation methods found in the literature.     

Modified solution of radial nozzle with thick reinforcement in spherical vessel head subjected to radial load

When nozzles in the spherical vessel head are designed by area replacement method defined in the majority of pressure vessel code, the thickness of the reinforcements is so thick that the application of the thin shell theory may not be appropriate in the shell assembly problems. To obtain the solution of the thick reinforcement in the radial nozzle of the spherical vessel head subjected to radial load, the nature of the thick shell is introduced to the current solution for the thin shell: (1) use of the meridional moment at the junction in moment equilibrium equation instead of the meridional moment at middle plane of reinforcement, (2) omission of derivative of rotation in the meridional moment equation, (3) omission of double derivative of radial displacement in meridional moment equation. The current analysis based on the thin shell theory is found to be less conservative as the thickness of reinforcement increases when the middle plane of the reinforcement does not coincide with the middle plane of the main shell of pressure vessel head. The rotation modified method addressed in (2) above is applicable to the design of radial nozzle reinforced by thick shell with t/R≥0.1 in spherical pressure vessel head subjected to radial load.     

A measurement of the Li(n,α) cross-section

The relative shape of the ⁶Li(n,α)/²³⁵U(n,f) cross-section ratio has been determined in the range of incident neutron energy from 2 to 800 keV. The measurements were made by the time-of-flight method using the Harwell 45 MeV linac to provide the pulsed source of neutrons. A thin ⁶Li-glass scintillator was used to register the (n,α) events, and the (n,f) events (in a metallic sample of ²³⁵U) were registered with fission neutron detectors. The shape of the ⁶Li(n,α) cross-section was obtained by combining the measured ⁶Li(n,α)/²³⁵U(tn,f) cross-section ratio with an evaluation of the ²³⁵U(n,f) cross-section. The cross-section so derived was placed on an absolute scale by normalization in the neutron energy interval 2–10 keV, where the ⁶Li(n,α) cross-section is accurately known. The cross-section at the peak of the prominent p-wave resonance near 240 keV is found to be 3.29 ± 0.12 b. The results are compared with other measurements and also with a recent theoretical calculation of the cross-section.     

Aircraft impact on a spherical shell

For nuclear power plants located in the immediate vicinity of cities and airports safeguarding against an accidental aircraft strike is important. Because of the complexity of such an aircraft crash the building is ordinarily designed for loading by an idealized dynamical load F(t), which follows from measurements (aircraft striking a rigid wall). The extent to which the elastic displacements of a structure influence the impact load F(t) is investigated in this paper. The aircraft is idealized by a linear mass-spring-dashpot combination which can easily be treated in computations and which can suffer elastic as well as plastic deformations. This ‘aircraft’ normally strikes a spherical shell at the apex. The time-dependent reactions of the shell as a function of the unknown impact load F(t) are expanded in terms of the normal modes, which are Legendre functions. The continuity condition at the impact point leads to an integral equation for F(t) which may be solved by Laplace transformation.F(t) is computed for hemispheres with several ratios of thickness to radius, several edge conditions and several ‘aircraft’ parameters. In all cases F(t) differs very little from that function obtained for the case of the aircraft striking a rigid wall. The calculation of the normal displacements w(t) at various points of the shell shows that the influence of the impact is bounded on a small region around the impact point. Therefore boundary conditions are not of interest. When calculating the normal displacement w(t) of the shell in consequence of the impact load F(t), we see a vibration of large amplitudes and of low frequency with a superposed oscillation of small amplitudes and a higher frequency. The fundamental frequency corresponds very well to that of the idealized aircraft, while the higher frequency belongs to a natural frequency of the hemispherical shell. It is of interest that not the fundamental natural frequency of the shell becomes visible but a higher one.If the ‘aircraft’ strikes a thin plate of similar parameters, the impact load is influenced more by the elastic deformation of the plate. Contrary to the hemisphere in this case only the two lowest eigenmodes are of interest. This different behaviour of shells and plates is investigated in the present paper. When calculating the amounts of energy of plates and these of shells with similar parameters we obtain very different values which show the different response of plates and of shells to impact load.     

Shape coexistence effects of super- and hyperdeformed configurations in rotating nuclei with 58 ≤ Z ≤ 74

Total energies are calculated for 138 even-even nuclei with proton numbers Z = 58 – 74 and for spins varying between I = 0 (ground states) and the maximum value corresponding to the fission limit induced by rotation. Calculations are based on the Strutinsky method and the cranking approximation. The results are presented in the form of two-dimensional maps of the total nuclear energy as a function of the quadrupole deformations, β₂ and γ, and spin. At each spin value and at each (β₂, γ) point the energies are minimized with respect to the hexadecapole deformation, β₄. The macroscopic energy term in the Strutinsky formula corresponds to the “folded-Yukawa plus exponential” formulation proposed by Møller and Nix. The microscopic energy term was calculated using the universal Woods-Saxon approach. Both terms were tested to ascertain that they reproduce the observed high-spin behavior of nuclei in the mass range considered.     

Strain-based finite elements for the analysis of cylinders with holes and normally intersecting cylinders

A finite element solution to the problems of stress distribution for cylindrical shells with circular and elliptical holes and also for normally intersecting thin elastic cylindrical shells is given in the present paper. Quadrilateral and triangular curved finite elements are used in the analysis. The elements are of a new class, based on simple independent generalised strain functions insofar as this is allowed by the compatibility equations. The elements also satisfy exactly the requirements of strain-free-rigid body displacements and uses only the external “geometrical” nodal degrees of freedom to avoid the difficulties associated with unnecessary internal degrees of freedom.A rectangular curved element was first developed and applied to the analysis of the familiar pinched cylinder and barrel vault problems (Ashwell and Sabir [1]). The results converge rapidly for displacements as well as for stresses. Further tests were carried out to investigate the ability of this element in predicting the high stresses in the neighbourhood of applied concentrated loads, (Sabir and Ashwell [2]). The loads considered were either radial or axial forces as well as moments about tangents to the circular cross section. The results obtained were not only in agreement with those of Forsberg and Flügge [3] but when plotted for the complex parameters defining proportions of the shell and flexibility as suggested by Calladine [4], their general forms corresponded closely with theoretical predictions.In the present paper we first develop strain based quadrilateral and triangular elements and apply them to the solution of the problem of stress concentrations in the neighbourhood of small and large circular and elliptical holes when the cylinders are subjected to a uniform axial tension. These results are compared with analytical solutions based on shallow shell approximations and show that the use of these strain based elements obviates the need for using an inordinately large number of elements.Normally intersecting cylinders are common configurations in structural components for nuclear reactor systems and design information for such configurations are generally lacking. The opportunity is taken in the present paper to provide a finite element solution to this problem. A method of substructing will be introduced to enable a solution to the large number of non banded set of simultaneous equations encountered. The solutions show good agreement when compared with experimental results of Corum, Bolt, Greenstreet and Gwaltney [5].     

MHD flow past a semi-infinite flat plate with heat transfer

Exact numerical solutions to the boundary layer similarity equations of MHD flow and heat transfer past a semi-infinite flat plate of an incompressible viscous fluid have been presented. The velocity of the fluid U and the magnetic field H₀ at a distance from the plate are both assumed to be uniform and parallel to the plate which is considered as isothermal. Velocity, magnetic and temperature fields have been shown graphically whereas the numerical values of ƒ′(0) and {−θ′(0)} are entered in tables. We observe that both ƒ′(0) and {−θ′(0)} decrease with increasing S (magnetic field parameter) and increase with increasing λ (ratio of magnetic diffusivity and viscous diffusivity).     

Break up time of fragmentating-solidifying UO2 spheres when quenched in sodium

When molten UO₂ is quenched in sodium, a sand-like debris results containing about 80% of fractured particles and 20% of smooth particles and spheres. The production of the fractured particles is normally explained by the thermal stress fragmentation model. Previously brittle fracture mechanics was applied to the complete solid shell of a freezing UO₂ drop, i.e. where 954°C < T < 2850°C; a calculation of fragmentation time was not possible. In this contribution the solid shell is continuously subdivided in a plastic or ductile layer for 1300°C < T < 2850°C and a brittle one for 945°C < T < 1300°C. Cracking occurs in the brittle layer only. In the present model a layer of a predescribed depth is assumed to ablate instantaneously, when the temperature reaches the transition point of elastic of ductile behavior (T = 1300°C) at its inner boundary. A new layer is formed within a time step, governed by the heat conduction equation. The discontinuous ablation process is thus related to the continuous progression of the solidification front. A calculation of the fragmentation time is possible: in principle it comprehends the summation of a large number of time steps for the formation of brittle layers. The thickness of the cracked brittle layer is parametrized to 20, 10, 5 and 1 μm. The concept of instantaneous ablation was suggested by the experience that the violent boiling forces of sodium are very effective on the UO₂ surface. The introduction of these minor changes makes the thermal stress model more realistic, because it can explain now, why UO₂ does not fragment in argon and water. The fragmentation time assessed for a UO₂ drop of 7.2 mm diameter in sodium, brittle layer 10 μm, is 250 ms.     

A 3-D Lagrangian stochastic model for the meso-scale atmospheric dispersion applications

A fully 3-D Lagrangian stochastic particle trajectory model is presented and applied to the meso-scale atmospheric dispersion and ground concentration calculations. The use of Gaussian plume model (GPM) with Pasquill–Gifford σ's for downwind distances exceeding 10 km is critically viewed. Further, the effect of variation in release height on the ground concentration and dispersion parameters (σ_y,σ_z) is studied for continuous releases. A continuous release of a non-buoyant gas in a neutral stratified atmosphere is simulated for various stack heights. The turbulent atmospheric parameters like vertical profiles of the fluctuating wind component and the eddy lifetimes for the horizontal and vertical directions, etc. were calculated using a semi-empirical mathematical model and compared with a E–ε model. The numerically calculated horizontal and vertical dispersion coefficients (σ_y,σ_z) are compared with the Pasquill–Gifford empirical σ's and with the Pasquill-modified σ_y. The ground concentration values as a function of downwind distance, have been compared with the Green Glow data and with a GPM for various release heights. The comparison of the results demonstrate a need of using a 3-D model over the simple GPM for meso-scale atmospheric dispersion applications. The GPM overpredicts the ground concentration because it cannot take into account the vertical wind shear, which is observed in the atmosphere under all stability conditions. A weak dependence on the release height in the numerically calculated dispersion coefficients σ's, is also observed.     

Measurements of thick-target (α, n) yields from light elements

High-precision measurements of thick-target yields from (α, n) reactions on Be, BeO, BN, C, UC, UO₂, Mg, Al, Si, Fe and stainless steel are described. The measurements cover an incident α-particle energy range of 3.6–10 MeV at energy intervals of 0.2 MeV in the energy range 4–7 MeV. The accuracy achieved is between 1 and 2% in most cases. As well as monitoring the incident α-particle energy by Rutherford scattering, precautions include a check for the presence of neutron-producing impurities in the targets by examination for γ-rays from (α, nγ), (α, pγ) and (α, α'γ) reactions.Values of the yields measured here from Be, Mg, Al and Si are 8–9% higher than recent published values. The measured yields in the compounds BeO and UC, when compared with those in Be and C, respectively, are used to assess a convenient approximate method of deriving thick-target yields in compounds from the values in the constituent elements.     

Comparative study of reinforced nozzle connections

The shell theory and finite-element method are used for the stress analysis of models of unreinforced and reinforced nozzle connections under internal pressure and external loads. Various reinforcement configurations are considered. The results of a comparative study of the effects of reinforcement on the maximum stress in radial models of the intersecting cylindrical shells are presented.     

Transient response of cylindrical shells to localized heat sources

An exact analysis of the dynamic problem on the thermal displacements and stresses of cylindrical shells subjected to instantaneous heating is carried out. Equations of motion for cylindrical shells, including the thermal effects, are reduced and solved by using the Fourier and Laplace transformation methods, after obtaining the temperature distribution determined from the nonstationary equation of heat conduction for a shell under a prescribed temperature field. Numerical results are presented for the thermal displacement and stresses due to the time variation together with the quasistatic ones.     

Kinetics modelling for three-body sequential reactions with applications to Li(n, n′)αT

An exact formalism is derived for the double differential cross section of a two-stage sequential reaction, which allows fully for the energy-angle correlations at each stage and for the lifetime of intermediate states. Application of the method is made to the two-stage process ⁷Li + n → ⁵He + ³H, ⁵He → ⁴He + n′ which contributes to the reaction ⁷Li(n, n′)αT. for the range of incident energies 5–14 MeV and with calculations of the contributions from the other major channel, ⁷Li + n → ⁷*Li + n′, ^7*Li → ⁴He + ³H, comparisons are made of calculated and measured values of dσ/dΩ and dσ/dE for ⁷Li(n, n′)αT. Whilst reasonable overall agreement is obtained, comparisons with the emitted neutron energy spectrum in the UKNDL and ENDF/BIV files indicate that there is considerably more structure present than is represented in either of these files.     

Fracture toughness of welded joints materials for main pipelines at Ignalina NPP

This paper deals with an investigation of mechanical and fracture toughness characteristics of welded joint materials used in Ignalina Nuclear Power Plant (NPP) reactor main circulating circuit (MCC) and steam pipelines. Basic metal of MCC group distributing header (GDH) steel 08Ch18N10T (Du-300), its weld metal welded by manual and automatic arc method using the wire SV-04Ch19N11M3 and electrodes EA-100/10U or EA-100/10T, this joint heat-affected zone metal and base metal of the main steam system—steel 16GS (DU-630) and its weld metal welded by manual arc method using the wire SV-08GS2 and electrodes UONI-13/55 were tested.Mechanical properties of welded joints materials—proportional limit (σ_pl), yield (σ_y) and ultimate (σ_u) strength, fracture stress (σ_f) and ductility (Z) (percent reduction of area) of the specimens were determined. Investigation of relative critical stress intensity factor for fixed thickness of the specimen and critical J-integral, J_IC, was performed.The probabilistic investigation of influence of the mechanical properties (σ_pl, σ_y, σ_u) onto fracture toughness characteristics and J_IC for tested materials by using linear regression model with three independent variables was performed.Research enabled to conclude that proposed multivariable regression model with 80% probability (confidence coefficient α = 0.05) has explained reasonably well the dependence of with σ_pl, σ_y, σ_u and it has shown the non-acceptability of probabilistic evaluation of the model with respect to J_IC.     

Numerical investigation of boiling regime transition mechanism by a Lattice–Boltzmann model

A numerical study has been performed to investigate the hydrodynamic aspects of the pool boilingon horizontal-, vertical- and downward-facing surfaces. The FlowLab code, which is based on a Lattice–Boltzmann (LB) model of two-phase flows, is employed. Macroscopic properties, such as surface tension (σ) and contact angle (β), are implemented through the fluid–fluid (G_σ) and fluid–solid (G_t) interaction potentials. The model is found to express a linear relation between the macroscopic properties (σ, β) and microscopic parameters (G_σ, G_t). The simulation results on bubble departure diameter appear to have the same parametric dependence as the empirical correlation. Hydrodynamic aspects of two-phase flow regime transition mechanism are investigated for different surface–coolant configurations. Results of the LB simulation clearly demonstrate that not only the bubble nucleation site density (related, e.g. to the heater surface condition and heat fluxes), but also the surface position have a profound effect on the flow regime (pool boiling) characteristics. The results of the LB simulation of hydrodynamics of two-phase flow on the horizontal surface provide the pictures quite similar to the experimental observation for saturated pool boiling. Two mechanisms of flow (boiling) regime transition on the vertical surface are predicted for the local bubble coalescence at bubble generation site and the downstream bubble coalescence. On the downward-facing surfaces, friction between bubbles and the surface wall is found to significantly enlarge the bubble size prior the bubble slip upwards. This behavior is responsible for the earlier bubble coalescence, and therefore, lowers the maximum heat removal rate, in a similar regime of nucleate boiling on a downward-facing surface.     

Iterative schemes for obtaining dominant alpha-modes of the neutron diffusion equation

Two new methods of obtaining dominant prompt alpha-modes (sometimes referred to as time-eigenfunctions) of the multigroup neutron diffusion equation are discussed. In the first of these, we initially compute the dominant K-eigenfunctions and K-eigenvalues (denoted by λ₁, λ₂, λ₃ … etc.; λ₁ being equal to the K_eff) for the given nuclear reactor model, by existing method based on sub-space iteration (SSI) which is an improved version of power iteration method. Subsequently, a uniformly distributed (positive or negative) 1/v absorber of sufficient concentration is added so as to make a particular eigenvalue λ_i equal to unity. This gives ith alpha-mode. This procedure is repeated to find all the required alpha-modes. In the second method, we solve the alpha-eigenvalue problem directly by SSI method. This is clearly possible for a sub-critical reactor for which the inverse of the dominant alpha-eigenvalues are also the largest in magnitude as required by the SSI method. Here, the procedure is made applicable even to a super-critical reactor by making the reactor model sub-critical by the addition of a 1/v absorber. Results of these calculations for a 3-D two group PHWR test-case are given. These results are validated against the results as obtained by a completely different approach based on Orthomin(1) algorithm published earlier. The direct method based on the sub-space iteration strategy is found to be a simple and reliable method for obtaining any number of alpha-modes. Also comments have been made on the relationship between fundamental α and k values.     

Thermal stresses in cylindrical shells based on improved theory

The thermal stresses in cylindrical shells subjected to local heating are studied by using the equations which include the effects of shear deformation. Developing the theory along lines identical to the Donnell-type formulation, the fundamental equation in terms of a single displacement containing the correction terms due to shear deformation and the thermal effects is found. The solutions for a finite cylindrical shell subjected to local heating are obtained using the Fourier transform method. The effects of the shear deformation on the thermal displacements and stresses are discussed by a comparison with the results due to the classical theory.     

Cross-sections of B(p,α)Be reaction for boron analysis

Nuclear reaction, ¹¹B(p,α)⁸Be, shows great potentials in boron characterization of materials due to its large cross-section at the broad resonance at 660 keV. Unfortunately, the existing cross-section data on this reaction show errors of up to 30% and inconsistency as high as 50%. All previously published cross-sections of this reaction were based on assumptions or on more or less arbitrary convention. We present an accurate measurement of the cross-sections of nuclear reaction ¹¹B(p,α₁)⁸Be^* using self-supported ¹¹B foil target, high resolution detection and careful analysis of the α-spectra. An accurate cross-section with α₁ was obtained with careful spectrum stripping of α₁ from the α₁₂ continuum. Cross-section with α₁+α₁₂ under more definite and realistic convention is also given for practical applications. Cross-sections are given in graphical form for θ_lab=150°.