Torispherical Pressure Vessel Research Articles

Buckling behaviour of auxetic domes under external pressure

In view of ongoing great interest in novel materials exhibiting Negative Poisson Ratio (NPR), the current study examines its influence on buckling behaviour of externally pressurized​ torispherical pressure vessel closures. Failure pressures are computed for the range of Poisson’s ratio: −1.0 ≤ν < 0.5. The following wall constructions are considered: (i) single auxetic layer, (ii) outer metallic skin supported on inside by an auxetic layer, and (iii) sandwich with auxetic core. Both metallic and auxetic layers are assumed to be isotropic, and exhibit elastic, perfectly plastic behaviour.Results of this numerical study show that the inclusion of auxetic can lead to 89% gain in failure pressures above auxetic-free configuration of the wall. The range of analysed diameter-to-wall thickness ratio varied between 666 and 2000. Two layer wall construction (metallic skin supported by auxetic layer), was found to exhibit better performance than the equivalent sandwich configuration.

Investigation of the Effects of Geometric and Load Perturbation to Buckling in Multilayered Torispherical Pressure Vessel Heads

The object of this paper is to investigate the effects of geometry and load perturbation to buckling in multilayered pressure vessel heads. The pressure vessel head in concern is thin walled torispherical geometry. Geometric and load perturbation can alter both the critical load for buckling and the buckled shape. Two and three layered torsispherical heads are considered. Two layered models include steel–aluminum and titanium–aluminum configurations and three layered models include copper–steel–copper configuration. Internally pressurized three-dimensional torispherical pressure vessel head model that is previously used in literature is constructed. As a first step eigenvalue solutions are obtained for each model. After this instability solutions with large deformation effects are conducted to obtain more realistic instability pressure values nonlinear. The solution is performed by finite element program ANSYS Workbench. In nonlinear analyses, perfectly plastic material model is used. It is concluded that geometric and load perturbations cause the instability pressure to decrease and cause the structure to buckle at a lower pressure value. It is also observed that for steel-aluminum configuration geometric perturbation is more critical than load perturbation whereas for aluminum-titanium the reverse is valid.

Investigation of the effects of perturbation forces to buckling in internally pressurized torispherical pressure vessel heads

The object of this paper is to investigate the effects of perturbation forces to buckling in pressure vessel heads. The pressure vessel heads in concern are confined to torispherical geometry with thin walls. Perturbation forces can alter not only the critical load for buckling but the buckled shape as well. In this paper in addition to previously used perturbation model, three more different perturbation force configurations are applied to the knuckle of the vessel. Internally pressurized three-dimensional torispherical pressure vessel head model that is previously used in literature is constructed and finite element program ANSYS Workbench is used for the solutions. First of all eigenvalue solutions are performed for each model. Then nonlinear instability solutions are conducted to obtain more realistic instability pressure values. For the nonlinear analyses not only large deformation static analyses but also large deformation transient analyses are conducted. For nonlinear analyses, perfectly plastic material model is used. It is concluded that instability pressures obtained by transient analyses are closer to plastic pressure values by PWC and ASME TES criteria and perturbation models increase instability pressure and equivalent plastic strain values.

Design by analysis of ductile failure and buckling in torispherical pressure vessel heads

Thin shell torispherical pressure vessel heads are known to exhibit complex elastic–plastic deformation and buckling behaviour under static pressure. In pressure vessel Design by Analysis, the designer is required to assess both of these behaviour modes when specifying the allowable static load. The EN and ASME boiler and pressure vessel codes permit the use of inelastic analysis in design by analysis, known as the direct route in the EN Code. In this paper, plastic collapse or gross plastic deformation loads are evaluated for two sample torispherical heads by 2D and 3D FEA based on an elastic-perfectly plastic material model. Small and large deformation effects are considered in the 2D analyses and the effect of geometry and load perturbation are considered in the 3D analysis. The plastic load is determined by applying the ASME twice elastic slope criterion of plastic collapse and an alternative plastic criterion, the Plastic Work Curvature criterion. The formation of the gross plastic deformation mechanism in the models is considered in relation to the elastic–plastic buckling response of the vessels. It is concluded that in both cases, design is limited by formation of an axisymmetric gross plastic deformation in the knuckle of the vessels prior to formation of non-axisymmetric buckling modes.

Evaluating Plastic Loads in Torispherical Heads Using a New Criterion of Collapse

In ASME Design by Analysis, the plastic load of pressure vessels is established using the twice elastic slope criterion of plastic collapse. This is based on a characteristic load-deformation plot obtained by inelastic analysis. This study investigates an alternative plastic criteria based on plastic work dissipation where the ratio of plastic to total work is monitored. Two sample analyses of medium thickness torispherical pressure vessels are presented. Elastic perfectly plastic and strain hardening material models are considered in both small and large deformation analyses. The calculated plastic loads are assessed in comparison with experimental results from the literature.

Gross Plastic Deformation of Axisymmetric Pressure Vessel Heads

The gross plastic deformation and associated plastic loads of four axisymmetric torispherical pressure vessels are determined by two criteria of plastic collapse: the ASME twice elastic slope (TES) criterion and the recently proposed plastic work curvature (PWC) criterion. Finite element analysis was performed assuming small and large deformation theory and elastic-perfectly plastic and bilinear kinematic hardening material models. Two plastic collapse modes are identified: bending-dominated plastic collapse of the knuckle region in small deformation models and membrane-dominated plastic collapse of the cylinder or domed end in large deformation models. In both circumstances, the PWC criterion indicates that a plastic hinge bending mechanism leads to gross plastic deformation and is used as a parameter to identify the respective plastic loads. The results of the analyses also show that the PWC criterion leads to higher design loads for strain hardening structures than the TES criterion, as it takes account of the effect of strain hardening on the evolution of the gross plastic deformation mechanism.

Round Robin Calculations of Collapse Loads—A Torispherical Pressure Vessel Head With a Conical Transition

Round robin calculations of collapse loads for a pressure vessel were made by 16 teams in Japan. The model is composed of a cylinder and a torispherical head with a conical transition. The structure is an example in which the stress classifications specified in the ASME Code are not strictly applicable. The calculations were performed to clarify the issue of the evaluation procedure using the limit analysis method specified in the ASME Code, Sect. III, and to check the sensitivity of calculation models and programs. It is found that the stress in the knuckle region has certain characteristics of secondary stress, yet still dominates the collapse of the vessel. Using the limit analysis to prove the validity of stress classifications is recommended. The sensitivity of the calculation methods is not so significant. Therefore, it is concluded that the limit analysis can be used as a standard procedure in regulations.

Simplified method based on the deformation theory for structural limit analysis—II. Numerical application and investigation on mesh density

A simplified finite element method based on the deformation theory for structural limit analysis is applied to the estimation of the limit load of two widely used pressure vessel components. One is the nozzle-sphere intersection under internal pressure and radial load: the other is the torispherical pressure vessel head under internal pressure. The results are compared with existing solutions and the detailed elasto-plastic finite element analysis. It can be seen that this method gives remarkably accurate results for various configurations of the components analyzed. In the meantime, the effects of mesh density on the accuracy of the solution are also examined. The investigation shows that with quite coarse mesh this method can obtain adequately accurate results and further reduce the computation resource.

Effects of local departures from nominal dimensions on stresses in thin torispherical end closures

A detailed survey (1-4)† of the variations in crown curvature, knuckle curvature and thickness of a number of thin torispherical pressure vessel ends has shown that actual dimensions may differ significantly from nominal dimensions. Numerical stress analysis using the finite element method has been carried out to assess the effects of these shape imperfections on the stresses in some of the thin torispherical pressure vessel ends considered in reference (4). The results have been compared with those for the corresponding perfect ends. In several cases significant increases in stress levels are predicted due to thickness and especially curvature variations.

Round Robin Calculation of the Collapse Load for Torispherical Pressure Vessel Head with Conical Segnents

Round Robin Calculation of the Collapse Load for Torispherical Pressure Vessel Head with Conical Segnents

Imperfection sensitivity of elastic and elastic-plastic torispherical pressure vessel heads

The paper deals with the results of a systematic numerical investigation of the nonlinear elastic and elastic plastic load-carrying behaviour and imperfection sensitivity of torispherical pressure vessel heads under uniform external pressure. In particular, the presentation focuses on the qualitative and quantitative influence of the radius-to-thickness ratio R/t, the yield stress σ 0 and the magnitude of initial geometric imperfections (in the shape of the elastic bifurcation mode) on the elastic-plastic load-carrying behaviour. It is found that thinner shells are more sensitive to the value of the yield stress and the magnitude of initial geometric imperfections, but their load-carrying capacity, relative to the elastic bifurcation pressure, may also be significantly higher than that of thicker shells.

A method of estimating limit loads by iterative elastic analysis. III—Torispherical heads under internal pressure

A simple method of estimating limit loads using a sequence of elastic finite element analyses and the lower bound theorem, termed elastic compensation, is demonstrated on the problem of the estimation of the limit behaviour of torispherical pressure vessel heads. Two possible techniques of elastic compensation are discussed, one which has general application and one possibly specific to heads. The results are compared to a series of detailed elastic-plastic finite element analyses and to classical solutions.

Very thin torispherical pressure vessel ends: Actual versus nominal dimensions

The results of a detailed survey of the variations in crown curvature, knuckle curvature and thickness of seventeen nominally torispherical pressure vessel ends are presented and discussed. Six of the ends were of the ‘crown and segment’ construction; eleven were ‘pressed and spun’. Three diameters (51in, 81in and 108in) were studied, and three thickness to diameter ratios (0·002 4, 0·001 6, 0·001 2). The material in all cases was a higher proof stress austenitic stainless steel (BS 1501: Part 3: 1973, Grade 304 S65).

Very thin torispherical pressure vessel ends under internal pressure: Strains, deformations, and buckling behaviour

Test results are presented for seventeen stainless steel torispherical pressure vessel ends, of production quality, loaded to failure under internal pressure. The nominal thickness/diameter ratios covered were 0.00237, 0.00158, and 0.0019, and the series included both ‘pressed and spun’ (eleven) and ‘crown and segment’ (six) ends. Details of the elastic, post-yield and buckling behaviour are given, and, where appropriate, predictions and relevant data from other sources are included for comparison. The full set of results is discussed in detail.

Very thin torispherical pressure vessel ends under internal pressure: Test procedure and typical results

Very thin cylindrical pressure vessels with torispherical end-closures have been tested under internal pressure until buckles developed in the knuckles of the ends. These were prototype vessels in an austenitic stainless steel. The preparation of the ends and the closed test vessels is outlined, and the instrumentation, test installation, and test procedure are described. Results are given and discussed for three typical ends (diameters 54, 81, and 108in.; thickness to diameter ratios 0.00237, 0.00158, and 0.00119). These include measured thickness and curvature distributions, strain data and the derived elastic stress indices, and pole deflection measurements. Some details of the observed time-dependent plasticity (or ‘cold creep’) are given. Details of two types of buckle that developed eventually in the vessel ends are also reported.

Equivalent torispherical pressure vessel heads

Lower bounds to limit pressures for ellipsoidal and torispherical heads of cylindrical shells have been calculated using a numerical optimisation technique. These have been used to explore the ideas of equivalence between torispherical and ellipsoidal heads. The geometrical proportions of torispherical heads which have optimum limit pressures have been found and these are compared with the results of three geometrical criteria of ‘equivalence’.

A Numerical Method for the Limit Analysis of General Shells of Revolution : 2nd Report, Lower Bound Method

A lower bound method of limit analysis of shells of revolution is proposed. The shell is assumed composed of a rigid/perfectly plastic material obeying the Tresca yield condition. The proposed method uses a part of the correct yield surface for shells of revolution, and gives statically admissible solutions. Using this method, the limit loads for two problems are calculated: (1) pressure vessels with rigid circular plates and (2) torispherical pressure vessels. The calculated results are shown in figures and compared with solutions by other methods. Further, experiments are performed with the torispherical pressure vessels, and their results are compared with the calculated results.

Instability Analysis of Torispherical Pressure Vessel Heads with Triangular Thin-Shell Finite Elements

The elastic instability of an internally-pressurized cylindrical tank with a torispherical head is investigated using a triangular, doubly curved, thin-shell finite element. The formulation of the finite element, which is based upon cubic displacement functions and a modified principle of potential energy, is first described. Then, the element is verified by comparing numerical results for the linear, stable analysis to alternative solutions for the same problem. The subsequent instability investigation includes the solution of the linearized problem of equilibrium bifurcation, that is, of the classical eigenvalue problem, and a general nonlinear analysis, based on tracing the nonlinear load-displacement path. The critical pressure, obtained with use of the general nonlinear analysis, agrees closely with an experimental result as well as with a numerical solution stemming from an axisymmetric formulation.

Analysis of Torispherical Pressure Vessels

The stress analysis of a torispherical pressure vessel is studied with the aid of two asymptotic expansions, one valid for large toroidal segments and another valid for smaller knuckles. A simple design formula for the maximum stress in the toroidal segment is obtained, which is in close agreement with experimental and with direct numerical results. Nonlinear prestress effect is also included. Graphs are presented from which stress concentration can be easily determined for a wide variation of parameters. A simple condition that yields the optimum toroidal radius for prescribed spherical cap and cylinder geometries is presented. Also, a reasonable lower limit for the critical internal pressure at which wrinkles are formed along the circumference of the toroid is found.

Buckling of orthotropic torispherical pressure vessels

Asymmetric buckling analysis of orthotropic pressure vessels, subjected to uniform external pressure is carried out. Sanders' nonlinear thin shell theory is employed to derive the buckling equations. The prebuckling state is represented by a linear axisymmetric bending state. The differential equations governing the prebuckling and buckling states are solved by numerical integration method using fourth order Runge-Kutta scheme. Results are presented for cylindrical pressure vessels with: (1) spherical attachment (2) torispherical head. The effect of L/R ratio ( L and R are length and radius of cylinder) on the buckling behaviour is investigated.