Uniform Internal Pressure Research Articles

Elastic buckling of complete toroidal shells of elliptical cross-section subjected to uniform internal pressure

It was predicted recently that some complete toroidal shells of elliptical cross-section would buckle when subjected to internal pressure. As yet there is no experimental evidence for this, so two independent shell buckling programs (BOSOR and INCA) were employed to calculate the internal buckling pressures for some test cases. The agreement between the results of the two programs is very good, with both programs predicting that buckling occurs.Calculations were also carried out by using BOSOR on complete toroids having cross-sections in the form of prolate (k=a/b>1.0) ellipses. The ranges of the parameters were: R/b=4 and 10, b/t=50, 100 and 200, and 1.3<k<2.5. The shells were assumed to be perfect, made from steel and to behave elastically. The buckling pressures and circumferential wavenumbers are given in tabular form and some are plotted graphically. The deformed shapes of some typical cross-sections prior to buckling are also illustrated, along with the buckling modes.

Nonlinear Deformation Analysis of Air-Dome Membrane Structures.

The present study treats the nonlinear deformation characteristics of air-dome membrane structures under uniform internal pressure. An incremental load method based on a linear approximation or a mathematical programming method is applied to analyze the nonlinear deformation behaviors by minimizing the potential energy. As examples of numerical analyses, the deformation characteristics are shown for elliptic membrane structures with various kinds of ffber orientation angles, for a membrane structure with an initial deflection, and for cable-reinforced membrane structures.

Optimal design of a cylindrical shell loaded by internal pressure

The problem of the optimal design of a noncircular cylindrical shell loaded by uniform internal pressure is presented. As an optimization criterion the minimal ratio of structure mass to medium mass is taken into account. Solution constraints are geometrical and strength conditions. Numerical analysis is realized using the finite element method. Optimal shapes of steel and aluminium shells are determined.

Semianalytical Solutions to Griffith Fracture Under Variable Pressure

The stresses and displacements in the vicinity of a Griffith crack in a semiinfinite two-dimensional elastic medium subjected to a variable internal pressure are analyzed by polynomial approximation and Fourier transform, and semianalytical solutions to the governing differential equations are formulated. The variable internal pressure is approximated by a polynomial and solutions are then obtained by Fourier transform. The expressions for the components of stress and displacement due to the opening of a crack under an uniform internal pressure are derived as a special case example to illustrate the use of the derived solutions. They are in agreement with solutions derived by other methods published in the literature. However, it is very difficult, if not impossible, to obtain exact analytical solutions when the distribution of the variable internal pressure becomes more complex. The derived semianalytical solutions establish the basis for a more efficient algorithm to obtain numerical solutions in such cases.

Material compatibility and some understanding of the ball swaging process

For a ball swaging process, an attempt has been made to get a closed-form solution for in-plane deformation. Inner and outer rings and arm have been approximated as thick cylinders subjected to uniform internal pressure. Based upon plastic deformations, residual stresses are calculated for perfect elasto-plastic materials and values of retention torques have been obtained. When treated as thin cylinders, it is proved that for perfect elasto-plastic materials, the sufficient, but not necessary, simple inequality shown below, if satisfied, ensures that retention torque is positive: /spl sigma//sub y3//E/sub 3/>/spl sigma//sub y2//E/sub 2/>/spl sigma//sub y1//E/sub 1/ where /spl sigma//sub yi/ and E/sub i/ are yield stress and Young's Modulus of cylinder i. Cylinder 1 is the innermost cylinder. This has also been validated for thick cylinders, numerically; but not proven mathematically.

Elastic buckling of cone-cylinder intersection under localized circumferential compression

Conical shells are often joined to cylindrical shells as end closures, reducers or roofs. Under a variety of loading conditions, the intersection between the large end of a cone and a cylinder is subject to a large circumferential compressive force which can lead to its failure by buckling. The problem may be idealized as a cone-cylinder intersection under a radial inward ring load. This paper first investigates the elastic buckling strength of thin cone-cylinder intersections under a radial inward ring load and develops simple and accurate equations for the prediction of buckling mode and strength. The ability of ring-loaded intersections to conservatively represent intersections under a variety of other loading conditions for their buckling behaviour is then explored. The ring load idealization is shown to be generally conservative, but may become rather conservative for some loading conditions such as uniform internal pressure. The strength of cone-cylinder intersections under uniform internal pressure is examined in detail in the final part of the paper and approximate strength equations are also developed, as this loading condition is important for pressure vessel and piping applications.

Cone-Cylinder Intersection under Internal Pressure: Nonsymmetric Buckling

Large circumferential compressive membrane stresses arise at the intersection of the large end of a cone and a cylinder when subjected to uniform internal pressure. Such intersections may fail by axisymmetric collapse involving excessive axisymmetric inward deformations or by nonsymmetric buckling. This paper presents a detailed finite-element investigation into the nonsymmetric bifurcation buckling behavior and strength of these intersections. Buckling pressures of a wide range of shell geometries are found, leading to the establishment of accurate and simple buckling strength equations. These equations are then used, together with the strength equations for axisymmetric collapse developed in previous work, to define the range of shell geometries whose strength is governed by nonsymmetric buckling.

Fibre-reinforced laminated composite tubes with free ends under uniform internal pressure

The stress and deformation fields in a fibre-reinforced composite tube under uniform internal pressure are discussed in some detail. In the interior region far from the ends classical laminate theory delivers rather poor results and has to be adjusted to include effects due to lateral contraction and to curvature. In the region near the ends boundary conditions (here stress-free ends will be assumed) require more elaborate methods of calculation. The use of finite element methods may prove to be problematic because in some parts of the boundary region very large gradients are expected. The problem is particularly acute in lay-ups with angle-plies where 3D-elements would be needed. In the following study analytical solutions based on asymptotic approximations of the three-dimensional equations of linear elasticity for homogeneous orthotropic materials will be presented. One “small” parameter ε R characterising thin shell geometry and another “small” parameter ε G following from homogenised material properties of the shell structure and whose order of magnitude is comparable with ε R are used to derive asymptotically consistent approximate solutions according to the following pattern: The adjusted laminate theory leads to stress distributions in each ply of the laminated tube which do not satisfy zero stress boundary conditions at the stress-free ends. In terms of asymptotic theory this is a typical problem of singular perturbations and can be solved by considering boundary layers near the free ends where stress and deformation fields satisfy the boundary conditions and match conveniently with stress and deformation distributions calculated with the adjusted laminate theory in the interior zone. To derive boundary layer equations which are easy to handle analytically and still obtain fairly accurate results, we replace the laminated structure by its homogeneous orthotropic equivalent. The boundary layer solutions are then obtained following the main ideas developed by Sayir [(1985) Local Effects in the Analysis of Structures, Elsevier Science Publishers; (1986) Bull. Tech. Univ. Istanbul 39, 515–528]. We first study as a leading example a hypothetical cross-ply structure and concentrate on transverse shear and axial normal stress distributions across the wall thickness and along the axial direction in the boundary layer region. We show that all boundary conditions near the tube ends can be fulfilled by introducing first a “normal-stress” boundary layer starting at the tube end and extending in the axial direction, then a shorter “shear-stress” boundary layer contained in the previous one and finally two lateral boundary layers starting at the inner and outer surface of the tube wall inside the shear stress boundary layer. We compare analytical results obtained with the help of the asymptotic method described above with those obtained from three-dimensional finite-element calculations for each ply. Then we discuss two cases of symmetric angle-ply lay-ups corresponding to tubes actually tested in the laboratory by the second author and enhance similarities and differences with respect to the cross-ply lay-up. We also show how the interlaminar shear stresses which may cause delaminations and failure depend strongly on the lay-up.

Impulse response of cantilever cylindrical shells with discontinuity in thickness subjected to axisymmetric load

Cylindrical shells with discontinuity in the thickness and subjected to axisymmetric impulse loads have been analyzed. The impulse loads considered are uniform internal pressure and circular ring load at the tip. The semi-analytical finite element method is used for the analysis. A thin shell finite element with four degrees of freedom per node is used. Three shell models are analyzed for possible reduction of maximum displacements and maximum dynamic stresses by choosing the discontinuity at different locations. Numerical results are presented for clamped-free boundary condition and for various step-ratios in the thickness. The weight of the shell is kept constant for various step-ratios. It has been shown that the maximum displacements and maximum dynamic stresses decrease for certain types of thickness variations presented in this paper.

Modelling the creep of a pipe weld using Cosserat theory

A mathematical model is developed which describes the steady state creep in a welded pipe which is subjected to a constant uniaxial end load and/or uniform internal and external pressure. The model is based on the Cosserat theory of plates and shells and a generalisation of Norton's law. Both asymptotic and analytical solutions are found and the results reveal that bending and thinning of the pipe take place on different length scales.

Dynamic response of cylindrical shells with discontinuity in thickness subjected to axisymmetric load

Abstract Cylindrical shells with discontinuity in the thickness and subjected to harmonic axisymmetric loads have been analyzed. The loadings considered are a uniform internal pressure and a circular ring load at the mid-section. A semi-analytical finite element method is used for the analysis. A thin shell finite element with four degrees of freedom per node is used. Three shell models are analyzed for possible reduction of maximum displacement and maximum stress by choosing the discontinuity at different locations. Numerical results are presented for clamped-clamped end conditions and for various step-ratios in the thickness, for the first three models. The weight of the shell is kept constant for the various step-ratios. It is shown that the maximum displacement and maximum stress decrease for certain types of thickness variations as presented in this paper.

Impulse response of cylindrical shells with a discontinuity in the thickness subjected to an axisymmetric load

Cylindrical shells with a discontinuity in the thickness and subjected to axisymmetric impulse load have been analyzed. The loadings considered are a uniform internal pressure and a circular ring load at the mid-section. A semi-analytical finite element method is used for the analysis. A thin shell finite element with four degrees of freedom per node is used. Three shell models are analyzed to indicate possible reductions of maximum displacements and maximum dynamic stresses by choosing the discontinuity at different locations. Numerical results are presented for clamped-clamped boundary conditions and for various step-ratios in the thickness. The weight of the shell is kept constant for the various step-ratios. It is shown that the maximum displacements and maximum dynamic stresses decrease for a certain type of thickness variation.

Incremental stresses in loaded orthotropic circular membrane tubes—I. Theory

Starting from the equations of general elastic nonlinear membrane theory in intrinsic form, the equations governing the incremental state of stress in an orthotropic circular membrane tube are derived and discussed. The tube is initially subjected to uniform internal pressure and to longitudinal extension, which lead to large homogeneous deformation. Then some changes in loading and/or geometry are considered, e.g. an additional load is applied, the shape of the boundary is changed or a slit is formed in the membrane. These changes are regarded as small perturbations on the initial homogeneous state of stress. The general form as well as some simplified forms of the equations are presented, leading finally to a set of two equations in terms of two scalar potential functions. Two different variational formulations for the problem are also presented, each of which may serve as a basis for numerical treatment.

Dynamic response of cantilever cylindrical shells with discontinuity in thickness subjected to axisymmetric load

Cylindrical shells with discontinuity in the thickness and subjected to harmonic axisymmetric load are analyzed for a clamped-free end condition. The loadings considered are uniform internal pressure and circular ring load at the tip. The semi-analytical finite element method is used for the analysis. A thin shell finite element method is used for the analysis. A thin shell finite element with four degrees of freedom per node and two nodes per element is used. Three shell models are analyzed for possible reduction of maximum displacement and maximum stress by choosing the discontinuity at different locations. Numerical results are presented for the first three modes and for various step-ratios in the thickness. The weight of the shell is kept constant for various step-ratios. It is shown that the maximum displacement and maximum stress decreases for certain types of thickness variations.

Analysis of thick laminated anisotropic cylindrical shells using a refined shell theory

Bending analysis of arbitrarily laminated, anisotropic panels and closed cylinders using the mixed shear deformation theory proposed by the authors is presented. A set of equilibrium, transverse shear compatibility and boundary conditions are obtained by using the mixed variational principle proposed by Jing and Liao (1989, Int. J. Numer. Meth. Engng28, 2813–2827) with displacements and transverse shear as independent variables. The zig-zag type displacement is assumed together with piecewise parabolic transverse shear. The initial curvature effect is included in the strain-displacement relations, stress resultants and assumed transverse shear stresses. Two types of shell geometry, infinitely long cylindrical panels and closed cylinders of finite length, are employed in the numerical study. The cylindrical panels considered are subjected to a transversely sinusoidal loading, while closed cylinders are under an uniform internal pressure. Numerical results presented here are compared with exact three-dimensional elasticity solutions. From these comparisons, it is found that this mixed shear deformation theory can supply reasonably good results.

Stress analysis of a storage vessel in the form of a complete triaxial ellipsoid: hydrostatic effects

Closed-form expressions for membrane stress resultants in a liquid-filled triaxial ellipsoidal storage vessel are derived. Unlike the ellipsoid of revolution, the triaxial ellipsoid has three different semi-axes, and hence does not possess axi-symmetry, necessitating a somewhat different analysis approach to that normally adopted for shells of revolution in general. However, instead of treating the vessel as a shell of completely arbitrary shape, and hence pursuing the method of stress functions usually employed for such shells, advantage is taken of the affine relationship between the geometries of the general ellipsoid and the sphere, by deriving the stress resultants in the ellipsoid directly from those for an equivalently-loaded spherical shell, the solution to the latter problem being readily obtainable on the basis of the theory of nonsymmetrically loaded shells of revolution (for which the general method of stress functions need not be used, as a more straightforward procedure is applicable). While the technique of using an affine transformation (to deduce the stresses in a complex shell from a solution for a related but simpler shell) is itself well known, the closed-form results for hydrostatic pressure that are presented herein are new, being readily applicable to the design of storage vessels and water tanks. In storage-vessel design, where both hydrostatic pressure and uniform internal pressure may be equally important, the present results complement those for the latter (and simpler) loading which, for the vessel geometry in question, are already available in the literature.

Stress intensity factors and crack extension in a cracked pressurised cylinder

This paper considers the plane elastic problem corresponding to single or multiple radial cracks emanating from the internal boundary of a circular ring, under uniform external tension and internal pressure. The stress intensity factors are calculated by using the dual boundary element method with the J-integral technique. Accurate data are found for varying crack depths over a representative range of wall ratios for fracture mechanics applications to pressurised circular cylinders. The interaction of multiple cracks and crack extension are investigated in the case of an internal pressure loading condition. The analysis shows that, for a multi-cracked pressurised cylinder, it is sufficient to calculate the stress intensity of the main crack in isolation for the purposes of safety assessment.

Theory of the crystal structures of selenium and tellurium: The effect of generalized-gradient corrections to the local-density approximation.

The crystal structure and phase stability of selenium and tellurium have been investigated by total-energy calculations within the local-density approximation (LDA), without and including generalized-gradient corrections (GGC). We find that the LDA underestimates the equilibrium volume by as much as 15% and predicts a crystal structure that is much more isotropic than observed: the nearest-neighbor distances in the helical chains are increased, but the interchain distances are reduced. Adding generalized-gradient corrections to the LDA results in a crystal structure, equilibrium volume, and binding energy in better agreement with experiment. We show that the effect of the GGC's is to add a uniform internal pressure to the system, the local bonding properties at constant volume, however, are unchanged.

A griffith crack located asymmetrically in an infinitely long elastic strip

Of concern in this paper is the distribution of stress in the vicinity of a Griffith crack located in a strip of finite width parallel to the edges. The edges of the strip are free from tractions and the crack is opened out by the application of a uniform internal pressure on its faces. The crack is not necessarily situated on the centreline of the strip, and so the mode II stress intensity factor is not necessarily zero even though the loading is purely of mode I type. By means of Fourier transforms the problem is reduced to that of solving a pair of simultaneous singular integral equations of Cauchy type. The mode I and mode II stress intensity factors are computed and presented in tables and figures for various combinations of h/δ and °/a, where 2δ is the strip width, 2a is the crack length and h is the distance of the crack from the centreline of the strip.For large δ/a one can derive approximate expressions for stress intensity factors by retaining terms up to a certain order of a/δ, say, (a/°)6. A comparison between the numerical results obtained from the approximate expressions and those obtained from the direct numerical solution of singular integral equations shows that the approximate expressions are good only for h = 0 and perhaps for small values of h/δ. It is observed that the convergence of the numerical results increases with the increase in the value of (δ-h)/a. This suggests that one should obtain approximate formulae for the stress intensity factors by retaining terms of a certain order of a/(δ-h). It is obviously done for the case when h = 0. However, for nonzero values of h/δ, the problem does not seem to be an easy one.A limiting case of the strip problem described above is a half plane problem. The latter problem of determining the distribution of stress in the neighbourhood of a crack, subjected to a uniform internal pressure and situated in the half plane parallel to its boundary, has been studied in the literature independently. We deduce the formulae for the half plane problem from the formulae for the strip problem and compare our results with those in the literature.

Cone‐Cylinder Intersection under Pressure: Axisymmetric Failure

This paper presents a detailed finite‐element investigation into the axisymmetric failure behavior and strength of intersections of the large end of a cone and a cylinder subject to uniform internal pressure. The study considers shells of uniform thickness with cone‐apex half angles ranging from 10° to 85° and cylinder radius‐to‐thickness ratios from 10 to 1,000. Plastic limit loads from a small‐deflection elastic‐plastic analysis are first presented. A rationally based simple equation is then established, which approximates the finite‐element limit loads very closely and is suitable for future incorporation in pressure‐vessel design codes. The effect of large deflections on the load‐deflection behavior is examined extensively for the covered geometric parameters for three different yield stresses. An approximate equation to assess the beneficial effect of large deflections on failure strength is also developed.