The condition of shakedown is examined for torispherical heads. The reason for using plastic analysis is to account for the strengthening that heads experience when subjected to internal pressure. Cyclic pressures are considered up to an allowable burst pressure that is based on the membrane stresses of the spherical part of the head. To simulate a proof test before service cycling, cases when the applied pressure is higher for the first cycle are also included. A definition of shakedown is used that places the limit of twice the yield strength on a fatigue stress parameter range that is defined in the paper. The equivalent stress and plastic strain ranges are calculated for ten head thickness-to-spherical radius ratios. From these data, shakedown pressures are obtained as fractions of the allowable burst pressure. By giving bounds for isotropic and kinematic strain-hardening models, the results are made independent from specific cyclic material behavior. It is also shown that if an elastic, geometrically linear algorithm is used, which is unable to account for the strengthening, the fatigue stress parameter range is overestimated for the thinner heads.

This paper examines the calculated pressure at a tensile plastic instability of a pressure vessel and its relationship to burst test results. It is proposed that the instability pressure be accepted as an upper bound to the pressure at which a vessel bursts, and that a strength reduction factor be used to predict the burst. The paper also presents a suitable mathematical model for the calculation of the instability pressures for thin-walled axisymmetric vessels. The proposition is tested by applying the model to a pressurized diaphragm, four cylindrical shells, and two torispherical heads, for which experimental burst data are available. It is found that the ratio of the test burst pressure to the calculated pressure at the tensile plastic instability, expressed in percent, ranges from 71 to 96 percent. The highest ratio occurs for a pressurized diaphragm with no significant defects. The lowest ratios occur for cylindrical shells with longitudinal welds, suggesting that the presence of the welds had a detrimental effect on the burst strength. These results may be useful when designing a pressure vessel with respect to its ultimate strength.

Two failure modes are addressed for cylinder-cone junctions under internal or external pressure: axisymmetric yielding and low-cycle fatigue. If the junction fails to meet the failure criterion of any one of the two modes, then it must be strengthened by reinforcement. It is shown in the paper that the degree to which a junction is strengthened depends on the distribution of the reinforcement. A placement of reinforcement on the cylinder alone, leaving the actual connection between the cylinder and cone unreinforced, adds strength with regard to axisymmetric yielding, but may not strengthen the junction sufficiently with regard to low-cycle fatigue. This means that the junction may appear reinforced, but is not strengthened. It is pointed out that the design rules of Section VIII, Div. 1 of the ASME B & PV Code (1992) set the need for reinforcement according to the failure criterion of low-cycle fatigue, while the distribution of the reinforcement is guided by the criterion of axisymmetric yielding. There is no assurance that the reinforced junction will meet the failure criterion of low-cycle fatigue. This means that the safety margin on the number of allowed cycles is less than that which is expected and that the junction may be unfit for cyclic service. It is also shown in the paper that a reinforcement distribution that requires minimum thicknesses for sections of both the cylinder and cone near the junction can satisfy criteria for both failure modes. This approach is already used in Code Case 2150 of Section VIII, Div. 1, for half-apex cone angles from 30 to 60 deg, and required in Div. 2 for cone angles from 0 to 30 deg. Its extension to angles from 0 to 60 deg for both internal and external pressure is recommended.

A theoretical model of an expanded tube-to-tubesheet joint is developed and examined with the objective to determine the residual stresses in the transition zone, which lies between the expanded and unexpanded regions of the tube. Owing to their effect on the development of stress corrosion cracks, the residual tensile stresses on the surfaces of the tube are of particular interest. A mathematical model that can predict these residual stresses is developed. Results of the model show that the maximum tensile residual stresses are axial and occur on the inside diameter of the expanded tube. It is shown in a parameter study that, for expansions that ensure a leak-tight joint, the maximum residual tensile axial stress on the inside surface of the tube reaches 80–95 percent of the yield stress of the tube, regardless of the geometrical and material parameters of the tube and tubesheet.

Hydraulic expansion of a tube into a grooved tubesheet is modeled as an elastic-plastic process. A special shell theory is used to obtain the tube residual stresses in the vicinity of the grooves. It is found that the maximum tensile residual stresses occur at the inside surface of the tube, but their magnitudes are lower than those in the transition region. The results also suggest the use of a higher hydraulic forming pressure for grooved joints than for ungrooved joints.

The interaction of a compliant slab and the flow of a viscous fluid is explored theoretically. Both laminar and turbulent boundary-layer flow cases are considered. The slab is treated as an infinite, elastic or viscoelastic solid of finite thickness, bonded to a rigid half-space. The two media are coupled through stresses and velocities, in both tangential and normal directions, on the deformed surface of the slab. The proposed mathematical model is used to predict the appearance of unstable surface waves of the coating. According to the model, instabilities of a highly viscoelastic slab originate in the form of waves with phase speeds at about 2.3 percent of the flow velocity while the wave speed for a slab with zero to moderate viscosity is in the range of 50–60 percent of the flow velocity, which confirms previous experimental results. The predicted onset velocities of slow propagating waves for turbulent flow over plastisol gel are within 10 percent of those observed experimentally.

The creep behavior in the crotch region of two normally intersecting cylindrical shells of equal diameter and thickness is treated with the aid of an axisymmetric shell model. The edge and pressure loads on this model, which has already proved to be useful in the case of elastic and elastic-plastic analysis, are chosen to simulate the loading in the crotch region. The model is used to calculate the redistribution of stress from the elastic state to the stationary state. By taking into account the wall thinning, which occurs as creep takes place, the critical elapsed time to rupture of a ductile vessel is estimated.

Stresses in the vicinity of the crotch of a pressurized, crossed cylinder pipe connection and a tee connection with equal diameter and thickness branches are determined by two different methods. In the first the structure is reduced to an equivalent, axisymmetric shell, while in the second, ovaling deformations are taken into account. The results agree well with each other and with published experimental results.

A procedure for the analysis of dynamic buckling of axisymmetric shells subjected to axisymmetric, periodic loads of long duration is proposed that is based on the calculation of the nonsymmetric modes of free vibration and associated mode integrals over the reference surface of the shell. Numerical results are presented for the evaluation of dynamic stability of an actual shell that is designed for the cooling system of a nuclear power plant.

A theoretical analysis for the determination of the contact pressure between a uniform elastic spherical shell and a rigid plate is developed. The results are directly applicable to the theory of applanation tonometry which is concerned with the measurement of intraocular pressure. Numerical results are presented for a shell having a radius to thickness ratio of 30.

The authors have previously shown that a thin, complete spherical shell compressed between two parallel rigid plates deforms initially with the polar portion of the shell flattened against the plates and that at a critical deformation the flat region may buckle into an axisymmetric inward dimple. The present paper presents an analysis of the stresses and deflections produced during axisymmetric postbuckling and determines the deformation states at which the shell may buckle into a nonsymmetric shape. The analysis accounts for finite deflections and rotations, but assumes that the material remains linearly elastic throughout the deformation. An experiment shows that both the primary axisymmetric bifurcation point and the secondary nonsymmetric bifurcation point are stable for a shell with R/h ≃ 40.

A procedure is given for the analysis of axisymmetrically imperfect spherical shells which is not limited by the magnitude of the imperfections. The geometric parameters of the imperfect shell are expressed in terms of those of the perfect shell and known imperfection distribution, and the imperfect shell is solved directly by means of a nonlinear theory. As an application of the proposed procedure, the critical pressures for an axisymmetrically imperfect complete spherical shell are calculated. The results are compared with those predicted by Koiter’s general theory of initial postbuckling behavior, and their asymptotic character is verified.

This paper presents the first phase in the analysis of the deformation of a shell of revolution when it is being crushed by a rigid wall. In this paper, the analysis accounts for finite deflections and rotations but assumes that the material remains linearly elastic. The load-deflection behavior is obtained for a hemispherical shell, and it shows that the shell cannot collapse with a flat contact region. The conditions for the buckling of the contact region are determined.

A multisegment method is developed for the solution of two-point boundary-value problems governed by a system of first-order ordinary nonlinear differential equations. By means of this method, rotationally symmetric shells of arbitrary shape under axisymmetric loads can be analyzed with any available nonlinear bending theory of shells. The basic equations required by the method are given for one particular theory of shells, and numerical examples of a shallow spherical cap and a complete torus subjected to external pressure are presented in detail. The main advantage of this method over the finite-difference approach is that the solution is obtained everywhere with uniform accuracy, and the iteration process with respect to the mesh size, which is required with the finite-difference method, is eliminated.

An exact solution is derived for the Green’s function of an open rotationally symmetric spherical shell subjected to any consistent boundary conditions. The fundamental singularity of the Green’s function is expanded in series according to the addition theorems of Legendre functions, and the solution for a spherical shell subjected to an arbitrarily situated, harmonically oscillating, normal, concentrated load is obtained explicitly in terms of associated Legendre functions. The corresponding static Green’s function is obtained simply by setting the driving frequency equal to zero. Numerical results for the displacements and stress resultants of an example are presented in detail.

This paper is concerned with fundamental solutions of static and dynamic linear inextensional theories of thin elastic plates. It is shown that the appropriate conditions which a fundamental singularity must satisfy at the pole follow from the requirement that the reciprocal theorem is satisfied everywhere in the region occupied by the plate. Furthermore, dynamic Green’s function for a plate bounded by two concentric circular boundaries is derived by means of the addition theorem of Bessel functions. The derived Green’s function represents the response of the plate to a harmonically oscillating normal concentrated load situated at an arbitrary point on the plate.