Parametric studies of cylindrical pressure vessels with different end closures

Recent developments in the finite element technique using incremental elements permit an easier and more precise determination of stresses, strains and displacements in cylindrical pressure vessels having different end closures. In this paper the geometry analysed is a cylindrical pressure vessel having hemispherical, torispherical, semiellipsoidal and toriconical heads. An axisymmetric solid finite element program employing incremental elements was used for a useful range of vessel parameters.

The formulation of the method used and the results of the parametric study obtained when internal pressure is applied to the vessel are presented. Results are reported in the form of stress intensity parameters based upon the mean circumferential stress of the cylindrical portion of the vessel away from the cylinder head junction.     

Elastic analysis of cylindrical pressure vessels with various end closures

The well-known problem of the elastic analysis of cylindrical pressure vessels with hemispherical, torispherical and ellipsoidal heads, involving the partial differential equations for the classical theory of thin shells of revolution axisymmetric in character, is attempted here using a step-by-step integration procedure and a segmentation technique. The numerical results are obtained with a generalised computer program for a number of cases and for a given set of values of elastic moduli, Poisson's ratio and $\text{thickness}\text{diameter}$ ratio. The results are compared with the known results available in literature and also with the stresses predicted by the ASME Code.     

An efficient design method for obround pressure vessels and their end closures

Obround cross sections find extensive industrial use in pressure vessels and pipings. Yet guidelines for their design are not available in the published literature. The obround shape is inherently unsuitable to withstand internal pressure. Support gussets must be added to such pressure vessels to render their use economical. This paper gives a method to efficiently design a pressurised obround cross section by determining the appropriate gusset plate locations, and the associated bending stresses. Formulae to design the obround flanged end closures are also given.     

Elasto-plastic analyses of cylindrical pressure vessels with reversed dished ends

Finite element stress analyses are presented which allow for elastic and elasto-plastic material behaviour for axisymmetric problems of pressure vessels using isoparametric quadrilateral elements with linear, parabolic and cubic displacements. The traditional method of analysis and design of cylindrical pressure vessels with reversed dished ends is based on the well known British Standard BS 5500 Code.¹ The stresses and displacements, especially at and near the junctions, are of great interest to designers. In view of the immense usefulness of such containers, a detailed post yield study has been made based on the stress-strain idealisation for isotropic strain hardening materials—H′= 0·0047E and Von Mises and Tresca yield criteria. The basis for the prediction of load increment sizes for the elasto-plastic analysis and the upper bound for collapse pressure are presented, as well as the excessive deformation and shakedown criteria suitable for the design of reversed dished end containers.     

Elasto-plastic analysis of a cracked ductile cylindrical pressure vessel

The deformation of a long cylindrical pressure vessel made of strain hardening material with a long longitudinal crack is analyzed under internal pressure by an FEM code. Stress distributions, the shape of the crack opening, the plastic zone, the J-integral and the CTOD are calculated.The $\text{J}\text{CTOD}$ ratio is found to be practically independent of the amount of pressure load, as well as of the crack height.Comparisons are made with a fully plastic strip yield model which seems to give a reasonable estimate of crack opening. This model suggests that the CTOD is linearly dependent on the radius of cylinder. This implies that the size of the pressure vessel is an important parameter when considering the tolerance of flaws.     

Stress analysis in cylindrical pressure vessels with loads applied to the free end of a nozzle

This paper contains a stress analysis of a cylindrical pressure vessel loaded by axial and transverse forces on the free end of a nozzle. The nozzle is placed such that the axis of the nozzle does not cross the axis of the cylindrical shell. The method of finite elements was applied to determine the state of stress in the cylindrical shell. The values obtained for stress in the nozzle region were used to determine the following: envelopes of maximum stress values; maximum values on these envelopes; and distances between maximum values on envelopes and the outer edge of the nozzle. Algebraic functions were determined, which enable easy and simple determination of these numerous stress values. The stress values obtained from the algebraic function were within −12.5 and +12.8% of those from finite elements. The difference between stresses deduced from strain gauge readings on experimental and calculated stresses was a maximum of 12%.     

Finite element analysis of cylindrical pressure vessels having a misalignment in a circumferential joint

Finite element analysis (FEA) has been carried out to obtain the elastic stress distribution at cylinder-to-cylinder junction in pressurized shell structures that have applications in space – vehicle design. To validate the finite element modeling and analysis results, three joint configurations, (viz., unfilleted butt joint with equal thickness, unfilleted butt joint with unequal thickness and filleted butt joint with equal thickness) having test results in open literature were considered. The peak stress values for these configurations obtained from FEA are close to that of test results. The peak stress value is found to reduce due to filleted butt joint as expected and also confirmed through test results.     

Fatigue failure criteria for thick-walled cylindrical pressure vessels

The fatigue failure of simple thick-walled cylinders under repeated internal pressure is considered with a view to establishing general criteria for failure which can be of use in design. The factors controlling the endurance limit and the limited life range are considered separately, utilising current thinking on the fatigue process. It is found that the fatigue behaviour of cylinders can be adequately predicted from conventional material fatigue data when the complexities of the elastic-plastic stress-strain state in a pressurised cylinder are taken into account. The importance of pressurising medium in assisting crack development is noted.     

Bursting pressure of mild steel cylindrical vessels

An accurate prediction of the burst pressure of cylindrical vessels is very important in the engineering design for the oil and gas industry. Some of the existing predictive equations are examined utilizing test data on different steel vessels. Faupel’s bursting pressure formula is found to be simple and reliable in predicting the burst strength of thick and thin-walled steel cylindrical vessels.     

Buckling under external pressure of cylinders with either torispherical or hemispherical end closures

In a number of applications, the actual boundary conditions at the ends of a cylinder are not taken into account properly when the structure is being designed against buckling. For example, in the design of submersibles the older theoretical treatments assume that bulkheads are present at the ends of the cylinders, whereas this form of construction is not always used.The purpose of the investigation described here is to study the effect of realistic boundary conditions on the elastic buckling pressure of unstiffened cylinders with torispherical or hemispherical end closures. In the present study only perfect, initially stress-free, structures are considered and their theoretical buckling pressures are obtained from a variational finite-difference program written for the digital computer.The numerical results presented were obtained from a limited parametric survey of the problem. In the main, linear buckling theory was used. However, as is shown, this can sometimes lead to unsafe predictions.The buckling pressures for the cylinders with hemispherical end closures, as predicted by the variational finite-difference technique, are also compared with a modified von Mises formula with corrections for the end closures. The agreement between the two sets of predictions was good within the range of the survey.     

Limit analysis of flush radial and oblique cylindrical nozzles in spherical pressure vessels. Part 2: Application of results in a design procedure

Using the method of a previously-published paper, optimised lower bounds to the limit pressure of a flush oblique cylindrical nozzle in a spherical pressure vessel have been obtained for a wide range of parameters and the results are presented in graphical form. The same technique of analysis and optimisation is applied to the case of a radial nozzle in a spherical pressure vessel. A very comprehensive parametric survey has been carried out and the results are again presented in graphical form. Methods of interpolation of both sets of results are also given.The use of the parameter is discussed as a means of simplifying the presentation of the results, and its limitations investigated. The results for the oblique nozzle are compared with previously-published experimental results.     

The design of thickness transition regions for GRP pressure vessels

A computer program (BOSOR4), which allows stress analysis of thin shells of linear elastic material, was used to investigate the design of thickness transitions between the end closure and cylindrical body of GRP pressure vessels. Attention is focussed on the junction of a hemispherical dome and cylinder. A simple geometry and tapered transition is proposed together with a simple but safe design procedure which gives a possible material saving of 16% for hemispherical domes compared with the recommendations of the GRP pressure vessel design code BS4994, 1973.     

Plastic behaviour of oblique flush cylindrical nozzles in cylindrical pressure vessels: An experimental investigation

Results of seven tests on cylindrical pressure vessels with flush cylindrical nozzles are reported. Two of the test specimens had radial nozzles and the rest had nozzles oblique to a radial line but normal to the longitudinal axis of the vessel. The specimens were machined out of solid 8 in diameter aluminium alloy bar. Electrical resistance strain gauges were used to measure the distribution of strain. Deflections were measured using clock-gauges and displacement transducers.     

Plastic collapse pressures for conical ends of cylindrical pressure vessels and their relationship to design rules in two british standard specifications

A numerical method is used to calculate lower bounds to collapse loads for cone-cylinder intersections. The method incorporates Carroll's optimisation procedure³ as developed by Robinson.⁵ Current methods of designing cone intersections, used in BS 1515,¹ BS 5500,² are discussed in relation to the results.     

Ratcheting limit of flat end cylindrical shell connections under internal pressure

Progressive plastic deformation, or ratcheting, is a failure mechanism in ductile structures usually associated with non-proportional actions. In the case of cylindrical shell flat end connections, with or without opening, ratcheting may occur under the action of cyclically varying pressure from zero to a maximum value, i.e. for single action and, therefore, proportional loading. In this paper the ratcheting limits for this connection under this loading and for two different models are given, for an ideal-elastic ideal-plastic constitutive law, Mises' yield condition and associated flow law. The results are compared with each other and with the limits for global plastic deformation given in the literature.     

A computer program for nonlinear analysis of pressure vessels

This paper describes the nonlinear analysis of pressure vessels necessary for taking into account the large deformations that take place at the junctions of shells of different geometries. Specifically, a computer program has been developed, based on both the linear and nonlinear theories of shells, which obtains numerical solutions for the most commonly used types of pressure vessels, namely those with spherical, ellipsoidal or conical heads and also flat-end pressure vessels. A multisegment integration technique has been used to obtain the solutions of the governing equations. The computed solutions are found to be highly accurate when compared with the known results of simple shells, as no nonlinear analysis is reported in the literature on the shell junctions in pressure vessels.     

Local and global collapse pressure of longitudinally flawed pipes and cylindrical vessels

Limit loads can be calculated with the finite element method (FEM) for any component, defect geometry, and loading. FEM suggests that published long crack limit formulae for axial defects under-estimate the burst pressure for internal surface defects in thick pipes while limit loads are not conservative for deep cracks and for pressure loaded crack-faces. Very deep cracks have a residual strength, which is modelled by a global collapse load. These observations are combined to derive new analytical local and global collapse loads. The global collapse loads are close to FEM limit analyses for all crack dimensions.     

Limit loads of circumferentially flawed pipes and cylindrical vessels under internal pressure

Published limit load formulae for circumferential defects overestimate the burst pressure for penetrating defects in pipes by the factor two in the short crack limit, because they only consider axial stress. Therefore, a class of limit load solution is discussed which takes the triaxial state of stress into account. The solutions for pressure loaded crack faces are improved analytically. Primal–dual limit analysis with the finite element method is used to adjust all solutions to numerical results. Limit loads are obtained for circumferential cracks of all sizes in thick-walled cylinders.     

A lower bound analysis for the calculation of limit pressure for a thick spherical vessel with a radial cylindrical nozzle

Using a three-dimensional stress formulation a lower bound analysis is presented for evaluating the limit internal pressure for a pressure vessel consisting of a thick spherical shell and a thick radial cylindrical nozzle. The analysis—which deals directly with stresses and the material yield criterion—can, in general, be applied to vessels of any thickness. The stresses are expressed in terms of an independent set of variables and the Von Mises yield criterion is used. The limit pressure is optimised using a non-linear programming method. Computed results are presented for a limited range of geometric parameters and a comparison is made with thin shell results.