Axisymmetric Solid Elements Research Articles

Acoustic Wave Propagation in Air-Filled Pipes Using Finite Element Analysis

The major objective of this work is to develop an efficient Finite Element Analysis (FEA) procedure to simulate wave propagation in air-filled pipes accurately. The development of such a simulation technique is essential in the study of wave propagation in pipe networks such as oil and gas pipelines and urban water distribution networks. While numerical analysis using FEA seems superficially straight forward, this paper demonstrates that the element type and refinement used for acoustic FEA have a significant effect on the accuracy of the result achieved and the efficiency of the computation. In particular, it is shown that the well-known, better overall performance achieved with 3D solid hexahedral elements in comparison with 2D-type elements in most stress and thermal applications does not occur with acoustic analysis. In this paper, FEA models were developed taking into account the influence of element type and sizes using 2D-like and 3D element formulations, as well as linear and quadratic nodal interpolations. Different mesh sizes, ranging from large to very small acoustic wavelengths, were considered. The simulation scheme was verified using the Time of Flight approach to derive the predicted acoustic wave velocity which was compared with the true acoustic wave velocity, based on the input bulk modulus and density of air. For finite element sizes of the same order as acoustic wavelengths which correspond to acoustic frequencies between 1 kHz and 1 MHz, the errors associated with the predictions based on the 3D solid hexahedral acoustic elements were mostly greater than 15%. However, for the same element sizes, the errors associated with the predictions based on the 2D-like axisymmetric solid acoustic elements were mostly less than 2%. This indicates that the 2D-like axisymmetric solid acoustic elements are much more efficient than the 3D hexahedral acoustic elements in predicting acoustic wave propagation in air-filled pipes, as they give higher accuracies and are less computationally intensive. In most stress and thermal FEA, the 3D solid hexahedral elements are much more efficient than 2D-type elements. However, for acoustic FEA, the results show that 2D-like axisymmetric elements are much more efficient than 3D solid hexahedral elements.

Numerical Analysis on Delta-Ring Seal Structure of Ultra High Pressure Vessel Employed in Food Processing

Due to the characters of the ultra high pressure vessel employed in food processing, a seal structure was introduced in this study. Then numerical analysis was performed using the larger finite element stress analysis software ansys12.0 for the stresses of the seal structure under internal pressure. In order to solve the contact question of delta-ring seal structure by using face-face contact model, a 3-D axisymmetric solid element was employed to calculate the stresses of the connected location among delta-ring, blind cover and end cylinder, the mises stresses in this structure were analyzed. Thus, the distribution nephograms of the stress on the contact surface were obtained in different conditions. Full-field plastic deformation of seal structure was generated when the inner pressure was larger than 300Mpa. At last, the reason for residual deformation was analyzed.

STL Representation of Axisymmetric Solid Element Mesh

This paper develops an algorithm for extracting stereolithography (STL) model from axisymmetric solid finite element (FE) mesh, so as to use 3D physical prototype instead of computer graphics-based visualization to express the deformation result of FE simulation. The key of the algorithm is to recognize contour node pairs from an axisymmetric solid element mesh, which exactly demonstrates that the proposed methodology is independent of axisymmetric element configuration, that is to say, it is applicable for either triangular, quadrilateral, or mixed triangular and quadrilateral axisymmetric solid element mesh.

Simple and accurate four‐node axisymmetric element

AbstractBased on the assumed strain method, a simple four‐node axisymmetric solid element is introduced. The assumed strain field is carefully selected to preserve the correct rank of the element stiffness matrix and to achieve high accuracy. The strain field is developed in conjunction with orthogonal projections and no matrix inversions are needed. The coarse mesh accuracy in bending and in typical axisymmetric load cases is excellent even for nearly incompressible materials. The strain‐driven format obtained is well suited for materials with non‐linear stress–strain relations. Several numerical examples are presented where the excellent performance of the proposed simple element is verified. Copyright © 2006 John Wiley & Sons, Ltd.

Coupled temperature–displacement modelling of injection stretch-blow moulding of PET bottles using Buckley model

This study presents a fully coupled temperature–displacement finite element modelling of the injection stretch-blow moulding (ISBM) process of polyethylene terephthalate (PET) bottles using ABAQUS with a view to optimising the process conditions. A physically-based material model (Buckley model) was used to predict the mechanical behaviour of PET at temperatures slightly above its glass transition temperature. A model incorporating heat transfer between the stretch rod, the preform and the mould was built using axisymmetric solid elements. Extensive finite element analyses were carried out to predict the deformation, the distribution and history of strain and temperature during ISBM of a 20g–330ml bottle, which was made in an in situ test on a Sidel SB06 machine. Comparisons of numerical results with the measurements demonstrate that the model can satisfactorily model the sidewall thickness and material distributions. It is also shown that significant non-linear differentials exist in temperature and strain in both bottle thickness and length directions during the process. This justifies the employment of a volume approach to accurately predict the final mechanical properties of the bottles governed by the orientation and crystallinity which are highly temperature and strain dependent.

Variable-node axisymmetric solid element and its application to adaptive mesh refinement

This paper presents an effective application of a variable-node axisymmetric solid element designated as AQV (Axisymmetric Quadrilateral Variable-node element). The variable-node element with physical midside nodes helps to overcome some problems in connecting the different layer patterns on a quadrilateral mesh in the adaptive h-refinement. This element alleviates the necessity of imposing displacement constraints on irregular (hanging) nodes in order to enforce the inter-element compatibility. Therefore, the elements with variable mid-side nodes can be used effectively in the local mesh refinement for the axisymmetric structures which have stress concentrations. A modified Gaussian quadrature should be adopted to evaluate the stiffness matrices of the variable-node elements mainly because of the slope discontinuity of assumed displacement within the elements. Some numerical examples show the usefulness of variable-node axisymmetric elements in the practical application.

Modeling of weld residual stresses in core shroud structures

This paper presents a computational model to predict residual stresses in a girth weld (H4) of a BWR core shroud. The H4 weld is a multi-pass submerged-arc weld that joins two type 304 austenitic stainless steel cylinders. An axisymmetric solid element model was used to characterize the detailed evolution of residual stresses in the H4 weld. In the analysis, a series of advanced weld modeling techniques were used to address some specific welding-related issues, such as material melting/re-melting and history annihilation. In addition, a 3-D shell element analysis was performed to quantify specimen removal effects on residual stress measurements based on a sub-structural specimen from a core shroud. The predicted residual stresses in the H4 weld were used as the crack driving force for the subsequent analysis of stress corrosion cracking in the H4 weld. The crack growth behavior was investigated using an advanced finite element alternating method (FEAM). Stress intensity factors were calculated for both axisymmetric circumferential (360°) and circumferential surface cracks. The analysis results obtained from these studies shed light on the residual stress characteristics in core shroud weldments and the effects of residual stresses on stress corrosion cracking behavior.

SIF evaluation and stress analysis of drillstring threaded joints

In a study by Tafreshi and Dover [1–3] stress analysis of drillstring threaded joints under axial, bending and torsion loadings was carried out using the finite element method. Stress concentrations were determined on a variety of threaded joints in order to predict the fatigue life of these components. Here some of those results are presented in more detail. In this study it is shown that in the case of bending, axisymmetric solid elements with non-linear, asymmetric deformation with fourier interpolation can be employed which reduces the computational time and modelling in comparison with the full three-dimensional analysis. Quasi semi-elliptic circumferential internal and external cracks in tubes with the same dimensions of the pins or boxes of a series of joints are modelled and analysed using the J-integral method. The result of this fully three-dimensional crack analysis shows the same trend as experimental [3] and published results [4].

Stresses analyses of shell structure with large holes

The strength, deformation and buckling of a large engineering structure consisting of four ellipsoidal shells, two cylindrical shells with stiffening ribs and large holes, one conical shell and three pairs of large flanges under external pressure, self weight and heat sinks have been analysed by using two kinds of five different finite elements - four assumed displacement finite elements (shell element with curved surfaces, axisymmetric conical shell element with variable thickness, three dimensional eccentric beam element, axisymmetric solid revolutionary element) and an assumed stress hybrid element (a 3-dimensional special element developed by authors). The compatibility between different elements is enforced. The strength analyses of the top cover and the main vessel are described in the paper.

End reinforcement of composite cylinders for biaxial testing

A detailed theoretical investigation has been carried out into factors affecting external end reinforcement of GRP (glass-reinforced plastic) cylindrical specimens for biaxial tests using axisymmetric solid finite elements. The reinforcement method is intended as a general solution to a wide range of applied axial and hoop stresses in the third quadrant of the stress field. Among the parameters considered were the ratio of hoop-to-axial stress, material employed for reinforcement, thickness, length and taper angles of the reinforcement layers. The assessment of the effective control of high discontinuity stresses at the fixed ends by the end reinforcement was made with the help of the Tsai-Wu strain index distribution and decay length of the bending stresses. The paper also describes the preliminary experimental work to show the effectiveness of an external composite reinforcement method in controlling the strain distribution of GRP cylinders closed by rigid end flanges.

New axisymmetric solid elements based on an extended Hellinger-Reissner principle

4-node incompatible axisymmetric hybrid stress elements are developed by a modified rational approach based on an extended Hellinger-Reissner principle. The stress fields are derived by using the constraint conditions that the equilibrium equations of higher order stress terms are imposed in a variational sense through the additional incompatible displacements. The resulting elements enjoy good performance in bending problems and at the nearly incompressible limit. Several examples are given to demonstrate the performance of the new elements.

Stresses and strains in hemispherical GRP pressure vessels with radial cylindrical branch connections

Experimental strain distributions are presented for two 1 m diameter chopped strand mat glass-fibre-reinforced polyester hemispherical pressure vessels with 200 mm diameter radial branch connections. One vessel had local reinforcement around the branch. The results are compared with theoretical predictions from computer programs based on thin-shell theory (BOSOR4) and axisymmetric solid finite element analysis (PAFEC, ASAS). The finite element approach is shown to give good agreement with experimental results when detailed information of the shape of the branch hemisphere junction is provided. In-plane and through-thickness stresses and strains are examined in detail for internal pressure and a radially outward load applied to the end of the branch. Fine finite element meshes are necessary for accurate prediction of through-thickness stresses at the junction. Orthotropic material properties are considered. Delamination occurred at the junction due to through-thickness tensile stresses, but for internal pressure loading the final fracture was due to in-plane stresses.

Two‐dimensional elastoplasticity: Approximation by mixed finite elements

AbstractTwo‐dimensional elastoplastic problems are used to assess the performance of a family of mixed finite elements in the non‐linear regime. In particular, four‐node bilinear quadrilateral elements that exhibit high accuracy in coarse meshes are used to simulate plane stress, plane strain, axisymmetric and (shear‐deformable) plate bending problems. The excellent performance of these elements is demonstrated with a number of discriminating examples selected from the literature. In addition, agreement is observed in the limit load predicted by the axisymmetric solid and plate bending elements.

Modeling Rotating Shafts Using Axisymmetric Solid Finite Elements with Matrix Reduction

An axisymmetric harmonic finite element representation is used to calculate shaft lateral critical speeds and perform stability analysis. Unlike a beam element model, an axisymmetric solid element representation allows the actual rotor geometry to be modeled. A Fourier series representation allows the three-dimensional shaft geometry to be modeled in two dimensions by only considering the radial and axial coordinates. Thus, the degrees of freedom of this element type are different from the usual two translations and two rotations at each node associated with bending of a three-dimensional beam element. A required gyroscopic matrix is also presented for completeness in analysis of rotating shafts. A matrix reduction technique is used to reduce the size of the shaft mass, gyroscopic, and stiffness matrices by condensing out slave degrees of freedom in terms of the retained master degrees of freedom. The formulation is applied to various examples for verification and to investigate the effect of selection of different master degrees of freedom for this element type on the results.

Buckling and postbuckling analysis of thick composite cylindrical shells under hydrostatic pressure

Thick composite-layered shells under hydrostatic pressure are analyzed for buckling and postbuckling response using axisymmetric solid elements. The numerical approach is based on p-version finite elements in conjunction with appropriate trigonometric functions. Particular attention is given to the evaluation of interlaminar stresses in the postbuckling range. The postbuckling response is determined using an asymptotic approach. It is found that layered shells with Z (the modified Batdorf parameter) < 1000 are imperfection-sensitive with respect to local buckling; shells with relatively small radius to thickness ratios develop significant interlaminar shearing stresses, of the order of 50% of the buckling pressure, for buckling displacements of the order of one-tenth of the shell thickness. Together, the imperfection-sensitivity and the high inter-laminar stresses can precipitate failure before the classical critical pressures are reached. These factors must be considered along with the potential for interaction of local and overall buckling in optimally designed, stiffened shell configurations. Numerical analyses demonstrate the effectiveness of the p-version approach in achieving rapid convergence of the buckling pressures, the imperfection-sensitivity parameter and the magnitude of interlaminar stresses. Wherever possible, comparisons have been made with results available currently in the literature. It is found that for long shells ( Z> 1000), the Donnell type formulations can seriously underestimate the buckling pressures. The present formulation based on three-dimensional nonlinear elasticity offers an accurate, powerful and yet computationally effective means of analysis of thick shells.

Experimental and finite element simulation methods for rate‐dependent metal forming processes

AbstractAn experimental procedure and a finite element simulation method for rate‐dependent metal forming processes are developed. The development includes the formulation of a tangential stiffness matrix for an axisymmetric solid finite element with four node, eight degree of freedom, quadrilateral cross‐section. The formulation includes the effects of elasticity, viscoplasticity, temperature, strain rate and large strains. The solution procedure is based on a Newton‐Raphson incremental‐iterative method which solves the non‐linear equilibrium equations and gives temperatures and incremental stresses and strains. Three examples are studied. In example 1, finite element simulation for the upsetting of a cylindrical workpiece between two perfectly rough dies is performed and the results are compared with alternative finite element solutions. In examples 2 and 3, both experimental and finite element studies are performed for the upsetting of a cylindrical billet and the forging of a ball, respectively. Annealed aluminium 1100 workpieces are used in both examples. For the finite element analysis, uniaxial compression tests are first performed to provide the material properties. The tests generate elastic moduli and two sets of stress‐strain curves (quasi‐static and constant strain rate), which are used to establish a rate‐dependent material model for input. For both examples 2 and 3, comparisons between the experimental and finite element simulation results for the forming force vs. die displacement relations and also for the deformed configurations show good agreement. The versatility of finite element methods allows for displaying detailed knowledge of the metal forming process, such as the distributions of temperature rise, yield stress, effective stress, plastic strain, plastic strain rate, forming forces and deformed configurations, etc. at any instance during the forming process.

Behavior of Welded Steel Water‐Storage Tank

The performance of a recently constructed welded steel water‐storage tank, believed to be the largest of its kind in the world, is monitored for thermal and live‐load (internal pressure) strains during initial filling. The experimental investigation and data are presented, along with an evaluation of the performance of the tank. Also presented for comparison to the measured stresses are predicted hoop and meridional stress distributions for approximate theories of cylindrical shell behavior. A finite element model using axisymmetric solid elements is used to account for the effects of nonuniform wall thickness and to model base restraint. Differences in predicted and analytical stresses are discussed. Differences due to unknown base restraint are observed and explained. Other observed differences, generally indicating conservative design procedures, are not conclusively explained, but two possible explanations are presented, including residual fabrication stresses and systematic measurement errors.

P-Version axisymmetric solid element for heat conduction

A nine-node two-dimensional axisymmetric solid element is presented for linear steady state axisymmetric heat conduction where the temperature field in the ξ and η directions of the element can be of arbitrary polynomial orders p ξ and p η . This is accomplished by first constructing the one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in the ξ and η directions using Lagrange interpolating polynomials and then taking the product (sometimes also called tensor product) of these hierarchical one-dimensional approximation functions and the corresponding nodal variable operators. The resulting approximation functions and the nodal variables are hierarchical, i.e. the element approximation functions and the nodal variables corresponding to the polynomial orders p ξ and p η are a subset of those corresponding to the polynomial orders p ξ + 1 and p η + 1 and as a consequence the element matrices and the equivalent nodal heat vectors are hierarchical also. The formulation ensures C 0 continuity, i.e. the continuity of the temperature across the interelement boundaries is guaranteed. The weak formulation of the Fourier heat conduction equation in the cylindrical coordinate system r, z is constructed and the element properties are derived using the weak formulation (or the associated quadratic functional) and the hierarchical temperature approximation for the element. The element formulation permits isotropic as well as generally orthotropic material properties. The formulation allows distributed heat flux, convective boundaries and internal heat generation. Numerical examples are presented to demonstrate the accuracy, simplicity of modeling, faster convergence rate and overall superiority of the present formulation over existing isoparametric axisymmetric solid elements. For each example the results obtained from the present formulation are compared with available analytical solutions and h-models employing isoparametric elements. Convergence graphs showing the rates at which the errors are decreasing are presented for both h and p-models. The results obtained from the present formulation exhibit excellent accuracy, faster convergence rate and require extremely simple models.

P-version hierarchical axisymmetric shell element

A hierarchical formulation for a three-node axisymmetric shell element based on p-version for linear elastic axisymmetric stress analysis is presented. The element displacement field can be of arbitrary polynomial orders p ξ and p η in the longitudinal (ξ) and the transverse (η) directions of the element. The element approximation functions and the corresponding nodal variables are derived by first constructing the one-dimensional hierarchical approximation functions of order p ξ and p η and the corresponding nodal variables operators in the ξ and η directions and then taking their products (also known as tensor products). The element displacement approximation is hierarchical, i.e. the approximation functions and the nodal variables corresponding to the polynomial orders p ξ and p η are a subset of those corresponding to the polynomial orders ( p ξ + 1) and ( p η + 1). Since the displacement approximation is hierarchical, the resulting element stiffness matrix and the equivalent nodal load vectors are hierarchical also. The formulation ensures C 0 continuity. The element properties are derived using the principal of virtual work and the hierarchical element approximation. In developing the element properties all four components of the stress ( σ θ , σ r , τ rz , σ z ) and strain ( ε θ , ε r , γ rz , ε z ) are retained, i.e. the stress normal to the middle surface of the element is not neglected (an assumption commonly used in developing axisymmetric shell elements) in the present formulation. The element formulation also permits non-differentiable geometries (sharp corners). The stress concentrations at such locations can be calculated accurately due to the presence of a complete state of stress (and strain) and a non-differentiable geometry feature. Numerical examples are presented to demonstrate the accuracy, efficiency, modeling convenience and overall superiority of the present formulation as well as its applications to pressure vessel stress analysis. The results obtained from the present formulation are also compared with those available in the literature as well as with the analytical solutions. The numerical results from the h- models using isoparametric axisymmetric solid elements are also presented for comparison purposes

Modelling of rotors with axisymmetric solid harmonic elements

In rotating machinery analysis, the rotating shaft is typically modeled by a series of line or beam elements. These beam elements are formulated from classical beam theory, with a basic assumption that “plane sections remain plane” during bending. However, when the rotor has abrupt changes in the diameter, this assumption is not valid; local distortion occurs near the section change, resulting in an increase in the overall bending flexibility of the shaft. In this paper the bending stiffness and natural frequencies of several shaft geometries with stepped diameter changes are compared using several modelling assumptions. The comparison includes beam approximations with both transfer matrix and finite element approaches, and available experimental data. In addition, the conceptual approach of using an axisymmetric solid model of the shaft is reviewed. Results are included for a commercial four-node element, and for a cubic element developed for this work. The effective use of matrix reduction in conjunction with an axisymmetric model is shown. The importance of non-planar distortion near section changes is reviewed.