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For leak-before-break (LBB) assessment, an idealized through-wall crack (TWC) is typically postulated to determine the critical crack length of cracked piping. However, such an idealization in terms of crack shape can lead to underestimations of plastic limit pressure. Although many studies have been performed to obtain accurate limit load solutions for cracked straight pipes by considering realistic crack geometries, there is still a lack of information regarding slant TWC at elbow. Therefore, three-dimensional finite element (FE) models of an elbow considering the effects of slant TWC on plastic limit pressure are developed. The proposed FE model and analysis procedure were verified through comparisons to the existing solutions for idealized TWCs in elbow. On this basis, the effect of slant TWC on the plastic limit pressure is analyzed, and a closed-form solution of the plastic limit pressure is proposed, for an elbow containing a longitudinal or a circumferential through-wall crack.

Analysis of the deformation of cylindrical membranes is important for understanding various mechanical systems, including biological structures such as the incudostapedial joint in the middle ear. Although the deformations of long cylindrical membranes have been extensively investigated, studies of short cylindrical membranes have not been published. Here we present an analytical solution for the deformation of moderate-length and short cylindrical membranes using the Mooney-Rivlin form for the strain-energy function. The membrane profiles and the internal pressures fit well with a finite-element model. We show that both moderate-length and short membranes become unstable when passing a certain limit pressure, and this instability persists even for very short membranes. Furthermore, we show that increasing the initial length, the pre-stretch and the stiffness all tend to stabilize a short membrane. The results may furnish insight into the mechanical behaviour of the incudostapedial joint.

This paper compares systematic finite element results with reference stress based J estimation equations for pressurised axial surface and through-wall cracked cylinders with the radius-to-thickness ratio ranging from five up to seventy. It is found that, for the reference stress, the use of the Tresca limit pressure given in R6 provides satisfactory results for all cases considered; for both inner and external axial surface cracks; for both semi-elliptical and constant-depth axial surface cracks. These results are obtained for high strain hardening materials such as Type 304 stainless steel.

The ultimate bearing capacity of an isolated column was investigated by combining a state of stress with a failure mechanism of lateral expansion in a cylindrical cavity which reproduces the stone column installation in a purely cohesive soft soil. Some analytical and empirical methods were used to determine the limit pressure of a cylindrical cavity with the assumption of varied behaviour laws of the medium around the cavity. These models permitted the calculation of the limit pressure of an expanded cylindrical cavity from which the ultimate bearing capacity was derived. Using recorded in situ data from load tests performed on 13 isolated column configurations, the model parameters are identified. A comparison between experimental data and predictions generated from several models is presented.

The paper aims to investigate plastic limit pressure of spherical vessels of nonlinear combined isotropic/kinematic hardening materials. The Armstrong-Frederick kinematic hardening model is adopted and the Voce hardening law is incorporated for isotropic hardening behavior. Analytically, we extend sequential limit analysis to deal with combined isotropic/kinematic hardening materials. Further, exact solutions of plastic limit pressure were developed analytically by conducting both static and kinematic limit analysis. The onset of instability was also derived and solved iteratively by Newton's method. Numerically, elastic–plastic analysis is also performed by the commercial finite-element code ABAQUS incorporated with the user subroutine UMAT implemented with user materials of combined hardening. Finally, the problem formulation and the solution derivations presented here are validated by a very good agreement between the numerical results of exact solutions and the results of elastic–plastic finite-element analysis by ABAQUS.

The transition in wear regime is a tribological phenomenon observed in many materials, where a sudden change in wear regime (from moderate to severe) occurs when operating parameters (e.g. pressure or sliding velocity) are changed. However, this phenomenon is rarely explored in brake friction materials (BFMs). Severe wear regime for BFMs results in an increase in cost and leads to environmental issues due to the high amount of loss in material. This work aims to study the transition in BFM applying different levels of pressure, where a BFM was rubbed against a gray cast iron disc in a tribometer. The tribofilm deposited on the disc surface and the area ratio of contact plateaus of BFMs were analyzed. The coefficient of friction (COF) and the BFM's wear were also measured. The wear regime changed from moderate to severe at a limit pressure of 90 bar. At the severe wear regime, a sudden increase in the COF was observed, and the wear of the BFM was approximately 13 times higher than in the moderate wear regime. It was also seen a removal of the tribofilm deposited on the disc's surface and a reduction of contact plateaus over the BFM's surface.

This paper provides effects of reinforcement shape and area on plastic limit loads of branch junctions under internal pressure and in-plane/out-of-plane bending, via detailed three-dimensional finite element limit analysis assuming elastic-perfectly plastic material behaviour. It is found that reinforcement is most effective when (in-plane/out-of-plane) bending is applied to the branch pipe. When bending is applied to the run pipe, reinforcement is less effective when bending is applied to the branch pipe. The reinforcement effect is the least effective for internal pressure.

This paper provides net-section limit pressures and a reference stress based J estimation method for pipes with constant depth, internal axial surface cracks under internal pressure. Based on systematic small strain finite element (FE) limit analyses using elastic perfectly plastic materials, net-section limit pressures are firstly determined, and based on FE results, a closed-form limit pressure solution is proposed. Furthermore, based on the proposed limit pressure solution, a method to estimate elastic-plastic J is proposed based on the reference stress approach. When the reference stress is defined by the proposed (global) limit pressure, estimated J values based on the reference stress approach are overall slightly lower than FE results, implying that the method is non-conservative. By re-defining the reference using optimised reference loads, resulting J estimates agree well with FE results.

An elastic-plastic analytical solution is derived for an internally pressurized thick-walled spherical shell of an elastic linear-hardening material using a strain gradient plasticity theory. Closed-form expressions are obtained for the stress, strain and displacement components. The inner radius of the shell enters the solution with its own dimensional identity as well as in non-dimensional forms, unlike that in classical plasticity-based solutions. As a result, the current solution can capture the size effect at the micron scale. The classical plasticity-based solution of the same problem is shown to be a special case of the present solution. To further illustrate the capability of the newly derived solution, formulas and numerical results for the plastic limit pressure are provided. These results reveal that the load-carrying capacity of the shell indeed increases with decreasing inner radius at the micron scale. The current solution can be employed to construct improved expanding cavity models in inde...

By using the model S – T diagram of the FeRh alloy in the F–AF transition region and the experimental data, it is shown that for inducing the transition adiabatically over the whole alloy volume, it is necessary to apply a pressure equal to or larger than a limit pressure P ↑ l , which is well above the critical one at the same temperatures. Thus, for inducing the complete transition at the critical temperature 308 K, at the critical temperature of the reverse transition 315 K and at 348 K adiabatically, it is necessary to apply a pressure of ∼0.21, ∼0.35, and ∼1.0 GPa, respectively. At that, the expected values of the alloy temperature change are 11.7, 12.2, and 15.0 K, respectively. It is shown that the difference in the temperature of ∼27 K near room temperature between high and low heat reservoirs can be reached when organizing the heat transfer cycles based on adiabatic inducement of the AF↔F transition by pressure within the limits of 1.0 GPa.

The authors present a new method for establishing the relations between the primitive surfaces and the teeth basic surfaces in a gear transmission. This new approach allows to make the synthesis not only by considering as known the gear pitch radius, or what is the same, its transverse module, as existing approaches, but also from the normal module in the pitch mean cone or the pinion pitch radius. There are two main factors that affect the size of the gears: the limit pressure angle and the mean tooth curvature or mean radius of the head cutter. This method allows to study the influence of the tooth normal module on the limit pressure angle, and the valid range of mean radius of the head cutter, according to the recommendation of Wildhaber, and their influence on the size of gears.

A general plane problem of the impression of a flat punch into a rigid-plastic half-space under the action of transverse and longitudinal shear contact stresses is considered. The condition of complete plasticity and the hyperbolic equations of the general plane problem of the theory of ideal plasticity [1] are used. The reduction of the limit pressure on the punch is determined as a function of the shear contact stresses.

Tresca limit pressures for long cylindrical shells and complete spherical shells subjected to arbitrary pressure, and several approximations to the exact limit pressures for limited pressure ranges, are derived. The results are compared with those in Section III-Subsection NB and in Section VIII-Division 2 of the ASME B&PV Code. It is found that in Section VIII-Division 2 the formulas agree with the derived limit pressures and their approximations, but that in Section III-Subsection NB the formula for spherical shells is different from the derived approximation to the limit pressure. The length effect on the limit pressure is investigated for cylindrical shells with simply supported ends. A geometric parameter that expresses the length effect is determined. A formula and its limit of validity are derived for an assessment of the length effect on the limit pressures.

Dome reversal in metal beverage containers refers to the event when the dome at the bottom of the containers snaps through due to overpressurization. This paper presents a combined experimental and numerical investigation of the problem. The response of the bottom of the containers considered is primarily axisymmetric prior to buckling and is characterized by a limit pressure instability. Under volume-controlled loading, collapse in an asymmetric mode occurs shortly after the limit pressure is achieved. An axisymmetric model of the problem was sufficient to calculate the limit pressure for the case considered. Results from a nonaxisymmetric model indicated that asymmetric modes became dominant after the limit pressure had been achieved.

In linear bearings there is contact over only small areas of the nominal contact area. The real contact pressure frequently is in the range of the plastic limit pressure. Regardless of the boundary lubrication, the asperity contact is mostly Hertzian-type contact. The developed contact algorithm can approximately simulate the elastic-plastic sliding contact behavior in the vicinity of the asperities by ignoring the effect of the tangential forces on the vertical displacement. Assuming certain sliding speed and coefficient of friction, the local temperature distribution is also calculated due to the heat generation over the real contact areas for slow sliding problems. In the described method, the original surfaces, having contact with an ideal smooth surface, produce mostly plastic contact, while the worn surfaces are mostly in elastic contact, having larger real contact areas. The sliding contact analysis of the original and worn surfaces shows significant differences in contact pressure distribution and heat generation.

Solutions for the expansion of cylindrical and spherical cavities in sands are presented. The sand is modelled using recently proposed critical-state models in which the values of the friction and dilation angles depend on the deformation history. Similarity solutions are obtained which enable the limit pressure to be calculated as a function of the initial conditions. Comparisons with existing perfectly plastic theories are made and consequences for the interpretation of cone penetrometer measurements are indicated.

The friction characteristics at the concentrated load area in a flat belt drive is investigated analytically and experimentally. It is observed that the friction force increases linearly in proportion to the normal force, up to the point where the limit pressure is equal to the maximum pressure and then remains almost constant for the normal forces, where the corresponding maximum pressure is greater than the limit pressure. The theoretical equation with limit pressure, developed in this paper, shows good agreement with the trend in the experimental results.

The investigation was concerned with a circumferential defect in a spherical shell at the outside surface adjacent to a protruding radial nozzle. The plastic limit pressure is important as one criterion of the two criteria method of safety assessment of such a vessel, and one objective was to compare the experimentally determined limit pressure with theoretical lower bound values available. Eight model vessels (two aluminium, six steel) with nominal spherical radius 69 mm, thickness 3 mm and nozzle mean radius 12·5 mm, thickness 3·5 mm, with machined, fully circumferential defects varying mainly in depth from zero to 0·75 of the vessel thickness, were tested under internal pressure. Three other vessels (one aluminium, two steel) were used to extend the information to part circumferential defects. The eleven vessels were finally tested to destruction under internal pressure and the results were compared with assessments which would be made using the two criteria method.

The article considers problems of finding the limit pressure leading to ductile failure of a cylinder with an axial crack. Actual formulas are obtained for a thin-walled shell and for cylinders with considerable thickness. The article shows their satisfactory agreement with the results of full-scale tests.

A limit analysis of a pad reinforced flush nozzle in a spherical pressure vessel has been published previously. A procedure for the design of a pad reinforced nozzle is proposed using the analysis and based on equating the limit pressure to the vessel test pressure. Design charts are presented for the case where the length of the reinforcing pad is that required to achieve the maximum limit pressure for a given set of dimensional parameters and also for the case where the length of the reinforcing pad is the minimum specified in present design codes. In all cases the amount of pad reinforcement required is greater than that for integral reinforcement using the same design principles.

Using a circumscribing yield surface, the limit pressure is obtained for a spherical shell with a circumferential partial penetration defect. The theory indicates the importance of the transverse shear stress for deep defects and is in good agreement with experiments on four vessels that are also described.

A cylinder-sphere pressure vessel, reinforced in the sphere by a section of constant thickness, has been analysed from the point of view of minimum weight. The reinforcement is allowed to be offset from the main sphere and the design has to be such that the test pressure of the vessel equals the limit pressure. It is shown that in most circumstances an economy of weight may be obtained by making the reinforcement thicker, but less extensive, than suggested in a previous proposal. Further benefit can be obtained by offsetting the reinforcement radially outwards so that the inside surfaces of main sphere and reinforcement are flush.

A lower bound to the limit pressure of a cylindrical shell with a rectangular slot part through the thickness at mid-length is computed using a linear optimization technique. The sides of the slot are axial and circumferential. The analysis has been carried out for internal pressure applied to the cylinder such that the longitudinal pressure end force is (a) not taken by the cylindrical shell, (b) wholly taken by the cylindrical shell. Different proportions of rectangular slot and different radius to thickness ratios of the cylinder have been considered. Results are presented for a range of parameters.

The limit pressure of a cylindrical shell with a two-dimensional array of penetrating nozzles and associated defects covering the whole shell has been calculated and compared with the limit pressure of the plain cylindrical shell. A lower-bound method was used. The conventional ligament efficiency method of calculating the limit pressure is too conservative. It is necessary to claim some strength from the nozzles. Hence a more elaborate method has been used here. If there are axial cracks in the main vessel tangential to some nozzles, the limit pressure is reduced to a value which is between the ligament efficiency and the value for a single crack in a plain cylinder. The actual reduction has been calculated for a range of crack lengths for a typical geometry.

The effectiveness of the length by which a radial branch pipe extends within a spherical pressure vessel is discussed with reference to the plastic limit pressure of the vessel and the design rules of BS 5500. Lower bound limit pressures are calculated using a non-linear programming method, the stress resultants being expressed in polynomial form. The material was assumed to be rigid-plastic and to obey the von Mises yield criterion.

This report discusses the results of a collaborative European Group on Fracture exercise to predict the pressure-crack size relationships of an experimental cracked pressure vessel which was undergoing tests for the CEGB. The details of the test and background mechanical properties data had been circulated to interested parties with a request for predictions to be returned prior to the vessel test results being published. The results of this exercise are described here, analysed in detail and compared with the test results. Collectively, eighteen individuals provided thirty eight different estimates of the vessel behaviour. The procedures used included R6 and variants of it, crack tip opening displacement approaches, engineering J design curve approaches, simple plastic collapse criteria and the GE J estimation scheme. Analysis of the predictions showed that: (1) in the prediction of maximum pressures, the plastic limit pressure was important; (2) in the prediction of initiation pressures the initiation toughness dominated; (3) the detailed form of the resistance curve had little influence on either the crack growth predictions or the maximum pressure predictions. (4) all forms of failure assessment diagram based upon the R6 axes were capable of producing accurate predictions. (5) where a poor model was used to represent the cracked section the results were unsatisfactory.

A method is proposed for calculating the lower bound limit pressure for thick shells of revolution under axisymmetric loading conditions. The method, which deals directly with stresses and the material yield criterion, can be applied to shells of any thickness. It is applied here to the problem of evaluating a lower bound to the limit pressure of a thick cylindrical vessel with torispherical ends of constant thickness. The stresses are expressed in terms of an independent set of variables and a non-linear programming method is used in the optimization of the limit pressure. Results are presented for a limited range of geometric parameters.

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 ϱ = ( a R )( R T ) 1 2 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.

A lower bound to the limit pressure of a flush oblique cylindrical branch in a spherical pressure vessel is computed using a non-linear programming method. The stress resultants in the branch and vessel are expressed in terms of an independent set of variables and the yield criterion used is one proposed by Ilyushin. Force/moment equilibrium at the junction of the branch and sphere is satisfied approximately by using one overal inequality constraint and the pressure is then optimized subject to the yield and equilibrium constraints. Results are presented for a range of geometrical parameters.

Using a one-moment limited interaction yield surface (Mф, Nф, Nθ) lower and upper bounds to the limit pressure have been calculated for pressure vessels consisting of the junction of a cylindrical and spherical shell. A full range of parameters (tT, rR, RT and tl) have been investigated extending from the case of a cylindrical flush nozzle in a spherical vessel to the case of a cylindrical vessel with hemispherical ends.

By using the results for a lower bound to the dimensionless limit pressure of radial and oblique nozzles in spherical pressure vessels given in Part 1 of this paper, a design procedure is proposed based on equating the limit pressure to the vessel test pressure. Some typical results are given, together with a discussion of the extent of reinforcement required in the branch and sphere. Using a published elastic analysis, the elastic stress concentration factors for the vessels designed by this method are presented.

Using a simple circumscribing yield surface for Nθ, N ф , M ф , the limit pressure has been calculated for a flush cylindrical nozzle in a spherical pressure vessel for a range of parameters r R , R T , t T . A lower bound and upper bound to the limit pressure have been derived using approximations similar to those used in other limit analyses of shells of revolution. In spite of the approximations, which are discussed, the analysis gives a useful insight into the plastic behaviour of nozzle/spherical shell intersections.

A complete solution is obtained for a rigid-plastic, cylinder-sphere intersecting shell subjected to internal pressure. The solution is presented in such a way that it enables one to compare the limit pressure, calculated on the basis of the uniform Tresca yield surface which is nonlinear, with the other linearized surfaces. It is shown that the limited-interaction yield surface, which has been used frequently, is not a good approximation for small diameter ratios, D/d and/or thin shell where T/D is small. Some experimental evidence is also presented in support of these conclusions.

Previously published work on the elastic and limit analysis of flush nozzles in spherical pressure vessels and on the effect of change of geometry on the limit pressure is considered. It is proposed that a suitable design procedure for pressure vessel nozzles is to identify a calculated lower bound to the limit pressure with the test pressure.

By restricting the stress system to one in which the principal stress directions are in the circumferential and longitudinal directions of the cylinder, a lower bound has been found to the limit pressure for a cylindrical pressure vessel with an unreinforced hole. The results are compared with experimental work on the limit pressure of cylindrical pressure vessels with branches.

The effects of the circumferential bending moment and change of circumferential curvature on the calculation of the limit pressure of symmetrically loaded shells of revolution are discussed with particular reference to a flush nozzle in a spherical pressure vessel. It is shown that if the circumferential bending moment is made zero in the equations of equilibrium, the resulting lower bound to the limit pressure is in fact the true limit pressure with only one restriction on the strain relationships.

Summary Twelve model pressure vessels with torispherical ends have been tested under internal pressure to investigate failure by instability and plastic deformation. The models covered combinations of three head heights and four thicknesses. All the thicker specimens with internal diameter-thickness ratios of 53, 106 and 212 failed by plastic deformation. The three thin specimens with an internal diameter-thickness ratio of 530 failed by buckling of the torus due to the circumferential compressive stresses. The experimental results for limit pressure and instability failure are compared with theoretical values. The effect of change of geometry is significant particularly for the larger head heights. For these specimens the experimental limit pressure is higher relative to theoretical predictions than is the case for the smaller head heights. A simple approximate theory is presented for predicting the pressure at which buckling occurs in the torus. The correlation of the new predictions with the experimental values is not good, but the new predictions are lower than those previously published.