Influence of flow stress choice on the plastic collapse estimation of axially cracked steam generator tubes

The influence of the choice of flow stress on the plastic collapse estimation of axially cracked steam generator (SG) tubes is considered. The plastic limit and collapse loads of thick-walled tubes with external axial semi-elliptical surface cracks are investigated by three-dimensional non-linear finite element (FE) analyses. The limit pressure solution as a function of the crack depth, length and tube geometry has been developed on the basis of extensive FE limit load analyses employing the elastic–perfectly plastic material behaviour and small strain theory. Unlike the existing solutions, the newly developed analytical approximation of the plastic limit pressure for thick-walled tubes is applicable to a wide range of crack dimensions. Further, the plastic collapse analysis with a real strain-hardening material model and a large deformation theory is performed and an analytical approximation for the estimation of the flow stress is proposed. Numerical results show that the flow stress, defined by some failure assessment diagram (FAD) methods, depends not only on the tube material, but also on the crack geometry. It is shown that the plastic collapse pressure results, in the case of deeper cracks obtained by using the flow stress as the average of the yield stress and the ultimate tensile strength, can become unsafe.     

Finite element-based limit load of piping branch junctions under combined loadings

The limit load is an important input parameter in engineering defect-assessment procedures and strength design. In the present work, a total of 100 different piping branch junction models for the limit load calculation were performed under combined internal pressure and moments in use of non-linear finite element (FE) method. Three different existing accumulation rules for limit load, i.e., linear equation, parabolic equation and quadratic equation were discussed on the basis of FE results. A novel limit load solution was developed based on detailed three-dimensional FE limit analyses which accommodated the geometrical parameter influence, together with analytical solutions based on equilibrium stress fields. Finally, six experimental results were provided to justify the presented equation. According to the FE limit analysis, limit load interaction of the piping tees under combined pressure and moments has a relationship with the geometrical parameters, especially with the diameter ratio d/D. The predicted limit loads from the presented formula are very close to the experimental data. The resulting limit load solution is given in a closed form, and thus can be easily used in practice.     

An approximative solution for limit load of piping branch junctions with circumferential crack and finite element validation

This paper is concerned with the prediction of limit load of the piping branch junctions with circumferential crack under internal pressure. Recently, we have developed a new approach for predicting the limit load of two-cylinder intersection structures with diameter ratio larger than 0.5, which has been successfully applied to defect free cases under various loading conditions. In the present work, we consider the extension of the approach to cover cracked piping branch junctions. On the basis of stress analysis in the vicinity of intersection line, a closed form of limit load solution for piping branch junctions with circumferential crack was developed. Then, 36 finite element (FE) models of piping branch junction with various dimensions of structure and crack were analyzed by using nonlinear finite element software. The limit loads from FE analysis and the proposed solution are compared with each other. Overall good agreement between the estimated solutions and the FE results provides confidence in the use of the proposed formulae for defect assessment of piping branch junctions in practice.     

Static strength analysis of CANDU-6 reactor fuel bundle

A static analysis, finite-element (FE) model was developed to simulate out-reactor fuel–string strength tests with use of the well-known, structural analysis computer code ABAQUS. The FE model takes into account the deflection of fuel elements, and stress and displacement in endplates subjected to hydraulic drag loads. It was adapted to the strength tests performed for CANFLEX 43-element bundles and the existing 37-element bundles. The FE model was found to be in good agreement with experiment results. With use of the FE model, the static behavior of the fuel bundle string, such as load transfer between ring elements, endplate rib effects, hydraulic drag load incurring plastic deformation in fuel string and hydraulic flow rate effects were investigated.     

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].     

Simplified J estimation methods for cracked cylinders under combined thermal and mechanical loading

In the field of non-linear fracture mechanics, a lot of work has been done on structures submitted to mechanical loadings. However, for thermal loadings and particularly for thermal shocks, only few results are available. The aim of this paper is to present the main results of a complete set of finite element (FE) computations, conducted by CEA, EDF and FRAMATOME, on cracked cylinders submitted to combined mechanical and thermal loads. The interaction between these two types of loads is analysed in the cases of austenitic and ferritic structures. Moreover, these results are compared to the predictions obtained by simplified engineering methods (R6 procedure and two French approaches) in order to improve them. Their domain of validity is also discussed.     

An alternative solution of the neutron diffusion equation in cylindrical symmetry

Alternative analytical solutions of the neutron diffusion equation for both infinite and finite cylinders of fissile material are formulated using the homotopy perturbation method. Zero flux boundary conditions are investigated on boundary as well as on extrapolated boundary. Numerical results are provided for one-speed fast neutrons in ²³⁵U. The results reveal that the homotopy perturbation method provides an accurate alternative to the Bessel function based solutions for these geometries.     

Optimized aspect ratios of restrained thick-wall cylinders by virtue of Poisson's ratio selection. Part two: Temperature application

Analytical and numerical modelling have been employed to show that the choice of Poisson's ratio is one of the principal design criteria in order to reduce thermal stress build-up in isotropic materials. The modelling procedures are all twofold; consisting of a solution to a steady-state heat conduction problem followed by a linear static solution. The models developed take the form of simplistic thick-wall cylinders such model systems are applicable at macro-structural and micro-structural levels as the underlining formulations are based on the classical theory of elasticity. Generally, the results show that the Poisson's ratio of the material has a greater effect on the magnitude of the principal stresses than the aspect ratio of the cylinders investigated. Constraining the outside of these models significantly increases the thermal stresses induced. The most significant and original finding presented is that the for both freely expanding and constrained thick-wall cylinders the optimum Poisson's ratio is minus unity.     

Plastic collapse assessment procedure for vessels with deep local thin area subjected to internal pressure

There are numerous cases in which the external surfaces of vessels such as towers, piping and storage tanks are partially corroded under their insulation. Some considerations on the plastic condition of cylinders with a local thin area have been reported, but a simplified evaluation procedure has not been established for the case in which a cylinder is simultaneously subjected to internal pressure and external bending moment due to earthquake, etc. Recently, the Ibaraki FFS rule has been presented, but the limitation on flaw depth (a) in the Ibaraki FFS rule is less than 0.7 (=a/t) of the original wall thickness (t). The effective cross section for the reference stress solution used to predict the plastic collapsed loads was investigated by nonlinear FEA. For the case in which a flaw depth to thickness ratio is large (a/t ≦ 0.7-0.95), the reference stress solution which decreases the effective cross section as a function of the flaw depth ratio was proposed.     

Evaluation of the finite-element-synthesis model using the 3-D finite-element technique

For the solution of multi group diffusion theory equations a 3-D finite-element (FE) code finerc has been developed. 3-D element shapes with the base orthogonal to the third direction are considered in finerc. For these elements the 3-D element submatrices of the FE formulation are easily computed in terms of the corresponding lower-dimensional element submatrices. The 3-D problems are tackled in finerc with a degree of complexity equivalent to that of 2-D problems.The 3-D FE technique is still somewhat expensive for routine design computations. The method can, however, be used for assessing the accuracy of other faster calculational methods. In this paper we have compared the results of the 3-D FE method with those of the FE-synthesis method which was previously developed by the present author. It is noted that the FE-synthesis method gives, at negligible computational cost, accurate eigenvalue estimates and reasonably good predictions of reactor core power profiles for the 3-D benchmark problems. The FE-synthesis method may be used for a number of survey-type analyses with occasional counter-checking by the 3-D FE technique.     

Elastic–plastic analysis of nuclear structures with millions of DOFs using the hierarchical domain decomposition method

The hierarchical domain decomposition method (HDDM) proposed by Comp. Sys. Eng. 4 (1993) 495 is applied to the large scale elastic–plastic finite element (FE) analysis of nuclear structures. The HDDM is a method to implement the finite element method (FEM) on various kinds of parallel environments. The substructure-based iterative methods can effectively be used with the HDDM to solve the large scale linear algebraic equations derived from the implicit FEM. In this paper, some key techniques to parallelize the static elastic–plastic FE analysis by the HDDM are described. As illustrative examples, a support structure of the high temperature engineering test reactor (HTTR), a pressure vessel, and an internal pump of a pressure vessel are analyzed. The structure of HTTR and the pressure vessel are modeled by hexahedral solid elements whose total degrees of freedom (DOFs) are about 1.3 millions (M) and 3 M, respectively. The internal pump is modeled by quadratic tetrahedral elements whose total DOFs are about 2 M. The elastic–plastic analysis of a simple cube with 10 M DOFs is also carried out. Both the conjugate gradient method for solving the linear equations and the Newton–Raphson method for solving nonlinear problems successfully converge.     

Pipe-anchor discontinuity analysis utilizing power series solutions, Bessel functions, and Fourier series

One of the paradigmatic classes of problems that frequently arise in piping stress analysis discipline is the effect of local stresses created by supports and restraints attachments. Over the past 20 years, concerns have been identified by both regulatory agencies in the nuclear power industry and others in the process and chemicals industries concerning the effect of various stiff clamping arrangements on the expected life of the pipe and its various piping components. In many of the commonly utilized geometries and arrangements of pipe clamps, the elasticity problem becomes the axisymmetric stress and deformation determination in a hollow cylinder (pipe) subjected to the appropriate boundary conditions and respective loads per se. One of the geometries that serve as a pipe anchor is comprised of two pipe clamps that are bolted tightly to the pipe and affixed to a modified shoe-type arrangement. The shoe is employed for the purpose of providing an immovable base that can be easily attached either by bolting or welding to a structural steel pipe rack.Over the past 50 years, the computational tools available to the piping analyst have changed dramatically and thereby have caused the implementation of solutions to the basic problems of elasticity to change likewise. The need to obtain closed form elasticity solutions, however, has always been a driving force in engineering. The employment of symbolic calculus that is currently available through numerous software packages makes closed form solutions very economical. This paper briefly traces the solutions over the past 50 years to a variety of axisymmetric stress problems involving hollow circular cylinders employing a Fourier series representation. In the present example, a properly chosen Fourier series represent the mathematical simulation of the imposed axial displacements on the outside diametrical surface. A general solution technique is introduced for the axisymmetric discontinuity stresses resulting from an anchor restraint on a selected of pipe geometry. These solutions can be economically implemented on today's symbolic calculus software packages with no loss in solution accuracy when compared to often more expensive techniques such as the finite element method. Verification of the axisymmetric solution technique is illustrated by the comparison of results for the closed form solutions versus those approximated by the finite element technique. Extensions of the general axisymmetric solution technique to other geometries and applied loads are also discussed while the numerical and graphical results are tendered.     

含整圈周向裂纹厚壁圆筒在内压和轴力共同作用下的极限载荷-简化滑移线场解

采用简化的滑移线场理论,推导出了含整圈环向裂纹厚壁圆筒在内压和轴力共同作用下的理想弹塑性材料极限载荷表达式,并采用有限元方法进行了验证.结果表明,对于含内表面裂纹的圆筒,其极限载荷解可由基于简化滑移线场的极限载荷解与无裂纹圆筒极限载荷解共同确定.有限元验证结果显示,理论解与有限元结果非常一致,且偏于保守.对含外表面裂纹的圆筒,其基于简化滑移线场的极限载荷解只适用于非常深的裂纹.其它情况建议使用基于Mises准则的解.     

Comparison of stress intensity factor solutions for cylinders with axial and circumferential cracks

Numerous stress intensity factor solutions have been proposed so far depending on the objects of evaluation including the variations of structures, cracks, and applied loads. In applying the flaw evaluation methodology to components of nuclear power plants, the use of reliable stress intensity factor solutions is essential. In this study, cracked cylinders were focused on as one of the typical configurations in actual plants. Existing stress intensity factor solutions for cracked cylinders were reviewed, and the accuracy of these solutions was investigated thorough the comparison with each other. Specific solutions were then recommended for cylindrical structures. Approximate expressions were newly derived for axially through-wall cracked cylinder subjected to linear stress distribution and for circumferentially through-wall cracked cylinder subjected to bending to realize simple evaluation of stress intensity factor. Considering that the cylindrical structures are often replaced with flat plates in the evaluation of actual components, the propriety of the replacement was also studied.     

Plastic collapse pressure of cylindrical vessels containing longitudinal surface cracks

Based on nonlinear finite element analysis, the plastic collapse pressures of cylindrical vessels with longitudinal surface cracks are computed. A general formula of plastic collapse pressure of such structures are given and compared with the literature solutions. The results of the present study could be applied for the integrity assessments, failure analyses, remanent life assessment, and licence extensions of the vessels.     

Natural convection heat transfer from horizontal rod bundles in liquid sodium. Part 1: Correlations for two parallel horizontal cylinders based on experimental and theoretical results

Natural convection heat transfer coefficients on two parallel horizontal test cylinders in liquid sodium were obtained experimentally and theoretically for various setting angles, γ, between vertical direction and the plane including both of these cylinders’ axes, over the range of 0°–90°. Both test cylinders are 7.6 mm in diameter and 50 mm in heated length with the ratio of the distance between each cylinder axis to the cylinder diameter, S/D, of 2. Theoretical equations for laminar natural convection heat transfer from the two horizontal cylinders were numerically solved for the same conditions as the experimental ones. The average Nusselt numbers Nu on the cylinders obtained experimentally were compared with the corresponding theoretical values on the Nu versus modified Rayleigh number R_f [= Gr*Pr²/(4?+?9Pr^1/2?+ 10Pr)] graph. The experimental values of Nu for the upper cylinder are about 20% lower than those for the lower cylinder at γ = 0° for the range of R_f tested here. The value of Nu for the upper cylinder becomes higher and approaches that for the lower cylinder with the increase in γ over the range of 0°–90°: the values for each cylinder agree with each other at γ = 90°. The values of Nu for the lower cylinder at each γ are almost in agreement with those for a single cylinder. The theoretical values of Nu on two cylinders except those for R_f < 4 at γ = 0° are in agreement with the experimental data at each γ with the deviations less than 15%. Correlations for two cylinders were obtained as functions of S/D and γ based on the theoretical solutions. A combined correlation for multi-cylinders in a vertical array based on the correlations for two cylinders was developed. The values by the correlation agree with the theoretical solution for the multi-cylinders for R_f ranging from 4.7 to 63 within 10% difference.     

Finite element mesh design of a cylindrical cask under puncture drop test conditions

Abstract

Transport casks for radioactive materials have to withstand the 9 m drop test, 1 m puncture drop test and dynamic crush test with regard to the mechanical requirements according to the IAEA regulations. The safety assessment of the package can be carried out on the basis of experimental investigations with prototypes or models of appropriate scale, calculations, by reference to previous satisfactory safety demonstrations of a sufficiently similar nature or a combination of these methods. Computational methods are increasingly used for the assessment of mechanical test scenarios. However, it must be guaranteed that the calculation methods provide reliable results. Important quality assurance measures at the Federal Institute for Materials Research and Testing are given concerning the preparation, run and evaluation of a numerical analysis with reference to the appropriate guidelines. Hence, a successful application of the finite element (FE) method requires a suitable mesh. An analysis of the 1 m puncture drop test using successively refined FE meshes was performed to find an acceptable mesh size and to study the mesh convergence using explicit dynamic FE codes. The FE model of the cask structure and the puncture bar is described. At the beginning a coarse mesh was created. Then this mesh was refined in two steps. In each step the size of the elements was bisected. The deformation of the mesh and the stresses were evaluated dependent on the mesh size. Finally, the results were extrapolated to an infinite fine mesh or the continuous body, respectively. The uncertainty of the numerical solution due to the discretisation of the continuous problem is given. A safety factor is discussed to account for the uncertainty.     

Estimation of fracture behavior of thin walled nuclear reactor fuel pins using Pin-Loading-Tension (PLT) test

For integrity assessment of fuel pins of nuclear reactors using fracture mechanics technique, the solutions for fracture mechanics parameters such as stress intensity factor (SIF), η and γ functions, J–R curve are necessary. These parameters are different for different pin geometries with various types of postulated flaws. These geometries do not conform to that of ASTM standard and hence, the above-mentioned parameters are usually not available. In this work, the authors have devised a pin-loaded tension (PLT) setup to determine the geometric functions for evaluation of stress intensity factors from experimental data using two different analytical methods. A 3D finite element (FE) analysis of the whole PLT setup has been carried out to evaluate the stress intensity factors and hence to validate the analytical solutions. FE analysis is carried out for various pin geometries (with different diameter and thickness values) to evaluate the limit loads and these solutions have been used to evaluate the η and γ functions which are essential for evaluation of the J-integral. Later, a load normalization technique has been used to estimate the J–R curves of a typical fuel pin geometry using the above η and γ functions. The results have been compared with available solutions from the literature.     

A simplified technique for shakedown limit load determination of a large square plate with a small central hole under cyclic biaxial loading

A simplified technique for determining the shakedown limit load of a structure was previously developed and successfully applied to benchmark shakedown problems involving uniaxial states of stress ( [Abdalla et al., 2007a], [Abdalla et al., 2007b] and [Abdalla et al., 2007c]). In this paper, the simplified technique is further developed to handle cyclic biaxial loading resulting in multi-axial states of stress within the large square plate with a small central hole problem. Two material models are adopted namely: an elastic-linear strain hardening material model obeying Ziegler's linear kinematic hardening (KH) rule and an elastic-perfectly-plastic (EPP) material model. The simplified technique utilizes the finite element (FE) method and employs small displacement formulation to determine the shakedown limit load without performing lengthy time consuming full elastic-plastic cyclic loading FE simulations or conventional iterative elastic techniques. The simplified technique is utilized to generate the shakedown domain for the plate problem subjected to cyclic biaxial tension along its edges. The outcomes of the simplified technique showed very good correlation with the results of analytical solutions as well as full elastic-plastic cyclic loading FE simulations. Material hardening showed no effect on the shakedown domain of the plate in comparison to employing EPP-material.     

Limit analysis and design of containment vessels

In the introduction, the theory of plastic analysis of shells is briefly recalled. Minimum-volume design for assigned load factor at plastic collapse is then considered and optimality criteria are derived for plates and shells of continuously varying or piecewise-constant thickness.In the first part, containers made of metal are examined. Analytical and numerical limit analysis solutions and corresponding experimental results are considered for various types of vessels, including intersecting shells. Attention is given to experimental post-yield behavior. Some tests up to fracture are discussed. New theoretical and experimental results of limit analysis of stiffened cylindrical vessels are presented, in which reinforcing rings are treated as discrete structural element (no smearing out) and due account is taken of their strong curvature. Cases of collapse by instability under internal pressure are pointed out. Minimum-volume design of circular plates and cylindrical shells is then formulated and various examples are presented of sandwich and solid metal structures. Containers of piecewise-constant thickness are given particular attention. Available experimental evidence on minimum-volume design of plates and shells is reviewed and commented upon.The second part deals with reinforced concrete vessels. Cylindrical containers are studied, from both points of view of limit analysis and of limit design with minimum volume of reinforcement. The practical use of the latter solutions is discussed.A third part reviews other loading cases (including cyclic and impact loads) and gives indications on corresponding theories, formulations and solution methods.The last part is devoted to a discussion of the limitations of the methods presented, within the frame of the “limit states” design philosophy, which is first briefly recalled. Considerations on further research in the field conclude the paper.