Thick Cylindrical Shells Research Articles

Applications of Iterative Elastic Techniques for Elastic-Plastic Analysis of Pressure Vessels

The paper presents an application of iterative elastic techniques for the determination of elasto-plastic stress-strain fields, limit and shakedown loads. The iterative elastic method relies on iterative modification of elastic modulus in regions at which elastically calculated effective stress exceeds material yield strength. The technique is applied first to thick spherical and cylindrical shells under combined pressure and thermal gradient. Results showed good correlation with analytical elasto-plastic solutions. The elastic compensation technique is then applied to predict elasto-plastic stresses and shakedown loads of thin spherical shell with cylindrical nozzle subjected to internal pressure or end-thrust loading. Predicted shakedown loads were found to be in good agreement with the well-known Leckie and Penny results adopted in pressure vessel codes.

Leaky wave scattering contributions for tilted thick cylindrical shells in water above the coincidence frequency: Ray theory and computational models

Scattering measurements [Morse, Marston, and Kaduchak, J. Acoust. Soc. Am. 103, 785–795 (1998)] and sonar images [Kaduchak, Wassmuth, and Loeffler, J. Acoust. Soc. Am. 100, 64–71 (1996)] demonstrate that above the coincidence frequency, the responses of tilted cylindrical shells associated with leaky antisymmetric Lamb waves can give major contributions to the scattering of sound. Thin-shell dynamics is not applicable for the modeling of the shell’s response and an approximate ray formulation has been developed for situations where exact solutions are unavailable. Relevant contributions include meridional [P. L. Marston, J. Acoust. Soc. Am. 102, 358–369 (1997)] as well as helical [P. L. Marston, J. Acoust. Soc. Am. 102, 1628–1638 (1997)] leaky rays. The present research emphasizes computational tests for situations where helical antisymmetric Lamb wave contributions are important. These include situations where contributions from the broad meridional ray enhancement interfere with helical ray contributions. The results are helpful for interpreting features of the scattering. [Work supported by the Office of Naval Research.]

Buckling design guideline for thin and thick cylindrical shells

Buckling design guideline for thin and thick cylindrical shells

Imperfection Sensitivity of Moderately Thick Laminated Cylindrical Shells

The instability problem of imperfect, laminated, moderately thick, circular cylindrical shells under the action of individual and combined application of uniform axial compression and external lateral pressure is investigated. The analysis is based on nonlinear kinematic relations, where the effect of transverse shear deformation is taken into account. The buckling is assumed to be elastic and the geometry to have initial geometric imperfections. A solution methodology has been developed and employed in generating results. The imperfection sensitivity is investigated. The results obtained indicate that geometric imperfections have small effect on the limit point load for moderately thick cylindrical shells. In addition, limited parametric studies have been made in order to assess the effect of length-to-radius ratio, radius-to-thickness ratio, and stacking sequence on the critical loads.

Localization of Buckling Patterns in Cylinders under Cyclic Loading

This paper investigates the cyclic-localization behavior of axisymmetric plastic buckling patterns of cylindrical shells in terms of its influence on strength deterioration. Since localization occurs due to a plastic bifurcation in the postpeak range, we first present a rigorous method to analyze the plastic bifurcation under the two-axial stress state, which is characteristic of the axisymmetric buckling. In this analysis, we also use an accurate cyclic plasticity constitutive model. The validity of the proposed theoretical approach to localization behavior is confirmed by a cyclic loading test on thick cylindrical shells. Then, using the proposed method, we precisely examine the effects of structural parameters as well as boundary conditions on localization behavior.

Temperature stresses in thick circular cylindrical shells used in nuclear reactors with and without uniform internal heat generation within the shell wall thickness

In the thick circular cylindrical steel shield of a nuclear reactor, the attenuated radiations generate heat which develops radial temperature profiles and consequent thermal stresses. In this first paper formulae and graphs are furnished which are used to calculate heat flow, temperature and stresses in the shell, away from external constraints, due to uniformly generated heat. Previous work and formulae [1] for the case where heat is not generated inside the thickness are also recalled. Triangular and exponentially generated heat will be examined in the following paper. Finally, the principle of superposition will be used to solve cases where the generated heat paths are not uniform, triangular or exponential.

Temperature stresses in thick circular cylindrical shells used in nuclear reactors with exponential heat generation within the shell wall thickness

In the preceding paper, formulae and graphs were given which enabled estimates to be made of heat flow, temperature and stresses through the thickness of the circular cylindrical shell, when heat was not generated inside the thickness and when it was generated according to a uniform path. Situations of heat generation in shell thickness are commonly encountered in the shields of nuclear reactors. All the expressions given for stress in the preceding paper may be reduced to the following form: σ{ Eα qa2/[(1 - ν) k]}η, where η is a factor associated with the stress which depends upon heat generation profiles, internal and external temperatures T a and T b of the shell and its thermal conductivity k, for each radius r. Another common heat generation which occurs in these shells has an exponential path. For these cases, the circumferential thermal stress σϑ may be given in the following form: σϑ = { Eα q(β−1)2/[(1 - ν) k]}ηϑ, where β is the exponential absorption factor which is related to the quantitative measure of the generated heat and ηϑ is the non-dimensional circumferential stress. In the present paper graphs are furnished to estimate the parameter ηϑ in order to foresee the values of stresses in the circumferential directions at the internal and external radii of the shell for the case of exponential heat generation through the wall thickness. They apply only when: (a) all generated heat flows inside the vessel or (b) all heat flows outside the vessel or ( c) T a = T b. Application of the principle of superposition shows how to evaluate the stresses where the generated heat paths are not uniform, triangular or exponential.

Nonlinear analysis of cross-ply thick cylindrical shells under axial compression

Buckling, postbuckling, nonlinear vibration and parametric resonance of thick circular cylindrical shells under axial compression are analyzed in this paper. The theory developed is based on a nonlinear and non-shallow thick shell theory, with its final equations involving two unknowns, the circumferential displacement v and the radial displacement w. The shell wall is cross-ply laminated. The plies are specially orthotropic, but the lamination can be unsymmetric. The axial load is assumed to be harmonically time dependent, or constant as a special case. The governing nonlinear partial differential equations are reduced to nonlinear ordinary differential equations in terms of time by the Galerkin procedure. Then, an asymptotic method is used to solve the resulting nonlinear ordinary differential equations. The numerical results for buckling loads are shown to compare very well with those of three-dimensional theories in the literature, even for very thick shells. The effects of lay-up and thickness on postbuckling equilibrium, nonlinear vibration and parametric resonance are demonstrated by examples.

On the stress concentration in thick cylindrical shells with an arbitrary cutout

In this paper, based on the theory of thick shells including effects of transverse shear deformations, a complex variables analytic method to solve stress concentrations in circular cylindrical shells with a small cutout is established. A general solution and expression satisfying the boundary conditions on the edge of arbitrary cutouts are obtained. The stress problem can be reduced to the solution of an infinite algebraic equation series, and can be normalized by means of this method. Numerical results for stress concentration factors of the shell with a small circular and elliptic cutout are presented.

A rigorous method for the analysis of localization of axisymmetric buckling patterns in thick cylindrical shells

This paper theoretically investigates the localization of axisymmetric buckling patterns in cylindrical shells because this phenomenon has a large influence on the strength degradation in the post-buckling range. The localization behavior is caused by a plastic bifurcation on the decreasing equilibrium path subsequent to the maximum load point. Thus, we first show a rigorous numerical method based on the finite-element method technique for analysis of plastic bifurcation under two-axial stresses. Then, using this method, we examine how the structural parameters, constitutive models and boundary conditions affect the deformation capacity of cylindrical shells.

Meridional and helical ray contributions to backscattering by tilted cylindrical shells: High-frequency tone burst and wide bandwidth measurements and interpretation

When viewed in the frequency-angle domain, the backscattering by truncated thick cylindrical shells in water exhibit various ridges of high backscatter. These enhancements are associated with leaky waves excited on the shell and can be described in terms of meridional and helical rays. The enhancements are not present for a rigid scatterer of the same shape and may be important for interpreting high-frequency sonar echoes. Previous work [Morse et al., J. Acoust. Soc. Am. (to be published)] has shown that these enhancements are not limited to near broadside incidence but can extend to end-on incidence in the coincidence frequency region. Results of broadband and tone burst measurements of these enhancements are presented. High-frequency tone bursts may be used to separate the meridional ray enhancement from the generally weaker helical ray enhancement. The general angular width of the meridional ray enhancement may be explained using an approximate ray theory [Marston, J. Acoust. Soc. Am. 102, 358–369 (1997)]. The amplitude of the backscattered meridional ray depends on the leaky wave reflection coefficient at the end of the cylinder which appears to be affected by mode conversion near mode thresholds. [Work sponsored by the Office of Naval Research.]

Elastodynamic Solution for the Thermal Shock Stresses in an Orthotropic Thick Cylindrical Shell

An elastodynamic solution for the thermal shock stresses in an orthotropic thick cylindrical shell is presented. The solution is achieved by the proper usage of integral transforms such as the finite Hankel transform and the Laplace transform. No restrictive assumptions on the shell thickness are placed. Results are presented for the well-formed wave propagation phenomenon of elastic stresses through the thickness of an orthotropic thick cylindrical shell. Thermal shock stresses become of significant magnitude due to stress wave propagation which is initiated at the boundaries by sudden thermal loading.

Koiter-Based Solution for the Initial Post-buckling Behavior of Moderately Thick Orthotropic and Shear Deformable Cylindrical Shells Under External Pressure

The initial post-buckling behavior of moderately thick orthotropic shear deformable cylindrical shells under external pressure is studied by means of Koiter’s general post-buckling theory. To this extent, the objective is the calculation of imperfection sensitivity by relating to the initial post-buckling behavior of the perfect structure, since it is generally recognized that the presence of small geometrical imperfections in some structures can lead to significant reductions in their buckling strengths. A shear deformation theory, which accounts for transverse shear strains and rotations about the normal to the shell midsurface, is employed to formulate the shell equations. The initial post-buckling analysis indicates that for several combinations and geometric dimensions, the shell under external pressure will be sensitive to small geometrical imperfections and may buckle at loads well below the bifurcation predictions for the perfect shell. On the other hand, there are extensive ranges of geometrical dimensions for which the shell is insensitive to imperfections, and, therefore it would exhibit stable post-critical behavior and have a load-carrying capacity beyond the bifurcation point. The range of imperfection sensitivity depends strongly on the material anisotropy, and also on the shell thickness and whether the end pressure loading is included or not. For example, for the circumferentially reinforced graphite/epoxy example case studied, it was found that the structure is not sensitive to imperfections for values of the Batdorf length parameter z˜ above ≃270, whereas for the axially reinforced case the structure is imperfection-sensitive even at the high range of length values; for the isotropic case, the structure is not sensitive to imperfections above z˜ ≃ 1000.

On propagation of shear crippling (kinkband) instability in a long imperfect laminated composite cylindrical shell under external pressure

A fully nonlinear analysis for prediction of shear crippling (kinkband) type propagating instability in long thick laminated composite cylindrical shells is presented. The primary accomplishment of the present investigation is prediction of equilibrium paths, which are often unstable, in the presence of interlaminar shear deformation, and which usually deviate from the classical lamination theory (CLT)-based equilibrium paths, representing global or structural level stability. A nonlinear finite element methodology, based on a three-dimensional hypothesis, known as layerwise linear displacement distribution theory (LLDT) and the total Lagrangian formulation, is developed to predict the aforementioned instability behavior of long laminated thick cylindrical shell type structures and evaluate failure modes when radial/hydrostatic compressive loads are applied. The most important computational feature is the successful implementation of an incremental displacement control scheme beyond the limit point to compute the unstable postbuckling path. A long (plane strain) thick laminated composite [90/0/90] imperfect cylinder is investigated with the objective of analytically studying its premature compressive failure behavior. Thickness effect (i.e. interlaminar shear/normal deformation) is clearly responsible for causing the appearance of limit point on the postbuckling equilibrium path, thus lowering the load carrying capability of the long composite cylinder, and localizing the failure pattern, which is associated with spontaneous breaking of the periodicity of classical or modal buckling patterns. In analogy to the phase transition phenomena, Maxwell construction is employed to (a) correct the unphysical negative slope of the computed equilibrium paths encountered in the case of thicker cylinders modeled by the finite elements methods that fall to include micro-structural defects, such as fiber waviness or misalignments, and (b) to compute the propagating pressure responsible for interlaminar shear crippling or kinkband type propagating instability. This type of instability triggered by the combined effect of interlaminar shear/normal deformation and geometric imperfections, such as fiber misalignment, appears to be one of the dominant compressive failure modes for moderately thick and thick cylinders with radiusto-thickness ratio below the corresponding critical value. A three-dimensional theory, such as the LLDT, is essential for capturing the interlaminar shear crippling type propagating instability.

High-frequency backscattering enhancements for thick truncated cylindrical shells in water at oblique incidence

Various authors have shown that the backscattering by a finite thin cylindrical shell at oblique incidence is enhanced when resonance conditions are met for the propagation of a leaky wave on the shell [see, e.g., M. L. Rumerman, J. Acoust. Soc. Am. 93, 55–65 (1993)]. To explore the backscattering enhancements relevant to high-frequency detection, impulse response measurements were carried out for slender thick cylindrical shells using a broad bandwidth PVDF sheet source, for tilt angles ranging from broadside to end-on incidence. These measurements reveal large backscattering enhancements beyond the shear wave cutoff angle, which extend to near end-on incidence in the coincidence frequency region. A simple approximation for the phase velocities of several surface waves is used to identify the underlying mechanisms. A broad enhancement feature is shown to result from the axial propagation of the supersonic a0 leaky Lamb wave. The associated meridional ray radiates a backward directed wavefront having a vanishing Gaussian curvature. Also observable are what appear to be resonance loci associated with the helical propagation of the subsonic a0−. [Work sponsored in part by the Office of Naval Research and by the ARL:UT Independent Research and Development Program.]

Green Functions for Axisymmetric Bending and Vibration of Thick Cylindrical Shells

It is shown herein that the axisymmetric bending and vibration problems of thick (Mindlin) cylindrical shells are analogous to the corresponding bending and vibration problems of Timoshenko beams resting on Winkler foundation. The Green functions for solving these two analogous problems are derived. Based on these Green functions, exact bending and vibration results may be obtained. The validity of the solutions has been verified via comparison studies with previous researchers' numerical results. Sample exact bending and vibration results are also presented for cylindrical shells in view to provide useful benchmark checks for researchers and engineers working on cylindrical shells and Timoshenko beams on elastic foundation.

Torsional instability of moderately thick composite cylindrical shells by various shell theories

Torsional instability of moderately thick composite cylindrical shells by various shell theories

Imperfection sensitivity of shear deformable moderately thick laminated cylindrical shells

The problem of instability of imperfect, laminated, circular cylindrical shells under the action of uniform axial compression is investigated. The analysis is based on nonlinear kinematic relations where the effect of transverse shear deformation is taken into account. The buckling is assumed to be elastic and the geometry to have initial geometric imperfections. The kinematic relations, governing equations and the related boundary conditions for the nonlinear analysis are derived and presented. A solution methodology has been developed and employed in generating results. The imperfection sensitivity is investigated. The results obtained indicate that geometric imperfections have little effect on the limit point load for moderately thick cylindrical shells.

Some simple solutions for buckling loads of thin and moderately thick cylindrical shells and panels made of laminated composite material

The objective of the present paper is to present a simple analytical solution for computing bifurcation buckling loads of thin and moderately thick orthotropic cylindrical shells and panels subjected to axial compression and normal pressure. That solution • — does not necessarily make use of the simplifying assumptions of shallow shell theory, and • — takes the flexibility with respect to transverse shear loads into account, which is supposed to be relatively high with laminated composite structures. The analysis is based on the governing nonlinear equations and boundary conditions of deep and shallow shell theory. They are linearised to yield the differential equations for bifurcation buckling. The linearised set admits a single-term analytical solution when the cylinder ends or panel edges are simply supported. It reduces the task to a small size (3∗3) algebraic eigenvalue problem that can be solved by standard methods. A computer program BOSCAP (Buckling of Orthotropic Shear-flexible Cylinders And Panels) is developed based on the above mentioned formulation which is very quick and easy to use during design trials or in automated optimization. A series of numerical examples is presented to demonstrate effects of transverse shear flexibility and curvature. Good agreement with some test results is shown.

WAVE DISPERSION MODELLING IN ANISOTROPIC SHELLS AND RODS BY THE FINITE ELEMENT METHOD

Harmonic wave propagation in shells and rods of infinite length is studied. Relations for a finite element model are derived in both Cartesian and cylindrical co-ordinates for the general anisotropy of a linearly elastic material. It is shown that the strain energy of a harmonic wave is not dependent on all elements of the elasticity matrix. Comparison of the proposed FEM solutions with analytical solutions, for both longitudinal and torsional waves in an isotropic cylindrical rod, shows excellent agreement. Some results for a thick cylindrical isotropic shell and an orthotropic square rod are also presented.