Energy-consistent theory of cylindrical shells

On the basis of the energy-consistent direction in the theory of shells, the deflected mode of circular cylindrical shells, considered as three-dimensional bodies, is studied. At that, two-dimensional equations of the boundary problem, obtained on the basis of the Lagrange principle by means of expansion of the sought relocations in polynomial series regarding the normal coordinate, are used. Equations are presented for the case when the approximation of the relocations in respect of the shell thickness retains summands that are by an order of magnitude higher than that in the Kirchhoff-Love classical theory of shells. The boundary problem formulated is solved with the help of the Laplace transform. As an example, a shell under the action of local loads is considered, for which the deflected mode is determined and a comparison is performed with the results obtained in accordance with the classical theory.     

On the creep buckling of circular cylindrical shells

Approaches to the creep buckling of clamped circular cylindrical shells subjected to axial compression combined with internal pressure are investigated with special emphasis on the concept of creep stability and the accuracy of the analysis. The instability in creep is discussed first, and it is elucidated that the pertinent phenomena can be analysed by a quasi-static method. Then, the formulation of the problem and the method of calculation of the resulting equations are studied. The influence of the creep laws and the non-linearity in field equations on the creep buckling of shells are also discussed.     

Elastoplastic buckling of axially compressed circular cylindrical shells

Buckling of axially compressed circular cylindrical shells is examined within the framework of small strain elastoplasticity. A linear bifurcation model is used along with the J₂ flow and deformation theories. All four possibilities of simple supports (SS1–SS4) and clamped edges (CL1–CL4) are considered. A general solution is given with special emphasis on the axially symmetric modes. Numerical results show that the weakening effect of the relaxed simple supports (SS1–SS2) is considerably reduced in the plastic range. A detailed comparison with available experimental data points in favour of the deformation theory.     

Optimally reinforced cutouts in laminated circular cylindrical shells

The problem of designing a cutout in a load bearing structural member in the form of a shell, such that the cut structure maintains its stress state with a minimal departure from the stress state of the uncut structure is addressed herein. Symmetrically laminated composite circular cylindrical shells under hydrostatic compression and axial pressure are considered. Shallow thin shell (Donnell shell theory) lamination theory is utilized. The original (uncut) stiffness of the shell structures is recovered considerably by appropriately designing an edge reinforcement around the cutout. The buckling load of the designed shells are analyzed via the finite element method. An experimental investigation has been carried out to verify some of the results obtained from the finite element analysis. In the work presented, the reinforcement is modeled as a one-dimensional rod/beam type structural element.     

Vibration and buckling of cross-ply laminated composite circular cylindrical shells according to a global higher-order theory

Natural frequencies and buckling stresses of cross-ply laminated composite circular cylindrical shells are analyzed by taking into account the effects of higher-order deformations such as transverse shear and normal deformations, and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for laminated composite circular cylindrical shells made of elastic and orthotropic materials is derived through Hamilton's principle. Several sets of truncated approximate higher-order theories are applied to solve the vibration and buckling problems of laminated composite circular cylindrical shells subjected to axial stresses. The total number of unknowns does not depend on the number of layers in any multilayered shells. In order to assure the accuracy of the present theory, convergence properties of the first natural frequency and corresponding buckling stress for the fundamental mode r=s=1 are examined in detail. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. Modal transverse shear and normal stresses can be calculated by integrating the three-dimensional equations of equilibrium in the thickness direction, and satisfying the continuity conditions at the interface between layers and stress boundary conditions at the external surfaces. It is noticed that the present global higher-order approximate theories can predict accurately the natural frequencies and buckling stresses of simply supported laminated composite circular cylindrical shells within small number of unknowns.     

Plastic analysis of filled, reinforced, circular cylindrical shells

Circular cylindrical shells made of anisotropic plastic material and filled with an incompressible fluid are subjected to a ring of load which is centrally located. The anisotropic material is a model of a composite material consisting of a ductile matrix reinforced either circumferentially or longitudinally with ductile fibers. Limit loads for all sizes of shell are obtained and their dependence on the degree of reinforcement is determined. The presence of the fluid considerably increases the limit loads over those for unfilled shells. The limitation due to finite transverse shearing stress is considered and is shown to have a significant effect on the strength of such structures.     

Elastoplastic dynamic instability of long circular cylindrical shells under pure bending

This paper presents a numerical study which is concerned with the prediction of the response and instabilities in long circular cylindrical shells under dynamic pure bending. Of particular interest is the response of such shells, bent into the plastic range of the material, and the various instability characteristics of the shells under dynamic bending (sudden step load). It was found that the major deformation characteristic of the shells is essentially similar to that observed in the static bending when the applied moment is much smaller than the critical dynamic moment. However, when the applied moment is close to the critical dynamic moment, the ovalization of the shell cross-section was found to be localized over a length of several shell diameters in the central region, even though the response of the shell curvature was shown to be still stable in this case. When the applied moment reaches the critical dynamic moment, the response of the shell curvature was shown significantly increasing with time and the shell buckled catastrophically. For thicker shells, it was found that the development of localized ovalization of the shell cross-section is the major factor that causes shell dynamic instability. For thinner shells, however, besides the localized ovalization, the bifurcation induced by short wavelength ripples on the compressed side of the shell was also observed in the initial buckling patterns. After the bifurcation, the initial buckling pattern was replaced by the final postbuckling mode characterized by a localized sharp cupping in the centre of the shell.     

Limit pressures for cylindrical shells with two adjacent circular cut outs

Cylindrical shells each with two equal circular openings were subjected to internal pressure until gross plastic deformation occurred. The opening diameter was kept constant throughout but the pitch was varied and two vessel sizes were used. In some specimens the two openings were disposed around the shell circumferentially and in some they were disposed along a cylinder generator. The experimental limit pressures so determined were compared with those calculated by using a lower bound analysis. Experimental limit pressures were also compared with those determined from tests on three specimens each having two adjacent openings with branch reinforcement.     

Vibrations of a cylindrical shell with partially constrained layer damping (CLD) treatment

This paper presents a mathematical model for a cylindrical shell with a partially constrained layer damping (CLD) treatment. A thin shell theory in conjunction with the Donnell–Mushtari–Vlasov assumptions is employed to yield the model. Employing the assumed-mode method, the discretized equations of motion in terms of shell’s transverse modal coordinates are derived. The effects of treatment length, of constraining layer (CL) thickness and stiffness, and of viscoelastic material core (VEM) thickness are then discussed. Numerical results show that thicker or stiffer CL warrants better damping. Thicker VEM does not always give better damping than thinner ones when CL exceeds a certain thickness.     

Displacements and stresses in cylindrical shells based on an improved theory

This paper is concerned with the study of the effect of shear deformation on the displacements and stresses in a cylindrical shell under a static load. Developing the theory including the effect of shear deformation, the system of equations with three displacements and two rotation angles is reduced to one with only three displacements, this being done without introducing any approximation. The accuracy of the theory proposed is examined for a simply supported cylindrical shell by comparing the results with those produced using the Mirsky-Herrmann type equations.     

Bifurcation modes in the limit of zero thickness of axially compressed circular cylindrical shell

Bifurcation intability modes of axially compressed circular cylindrical shell are investigated in the limit of zero thickness (i.e., h (thickness) → 0) analytically, adopting the general stability theory developed by Triantafyllidis and Kwon(1987) and Kwon(1992). The primary state of the shell is obtained in a closed form using the asymptotic technique, and then the straightforward bifurcation analysis is followed according to the general stability theory to obtain the bifurcation modes in the limit of zero thickness in a full analytical manner. Hence, the closed form bifurcation solution is obtained. Finally, the result is compared with the classical one.     

Nonlinear dynamic response of rotating circular cylindrical shells with precession of vibrating shape—Part I: Numerical solution

The nonlinear dynamic response of a cantilever rotating circular cylindrical shell subjected to a harmonic excitation about one of the lowest natural frequency, corresponding to mode (m=1, n=6),where m indicates the number of axial half-waves and n indicates the number of circumferential waves, is investigated by using numerical method in this paper. The factor of precession of vibrating shape ? is obtained, with damping accounted for. The equation of motion is derived by using the Donnell’s nonlinear shallow-shell theory, and is general in the sense that it includes damping, Coriolis force and large-amplitude shell motion effects. The problem is reduced to a system of ordinary differential equations by means of the Galerkin method. Three different mode expansions are studied for finding the proper one which is more contracted and accurate to investigate the principal mode (i.e., m=1, n=6) response. From the present investigation, it can be found that for principal mode resonant response, there are two traveling waves with different linear frequencies due to the effect of precession of vibrating shape of rotating circular cylindrical shells; the effects of additional modes n and k (multiples of frequency) on the principal mode resonant response are insignificant compared with an additional mode m, showing that it is better to adopt two neighboring axial modes to study the principal resonant response of the system.     

Vibration of functionally graded cylindrical shells

Functionally gradient materials (FGMs) have attracted much attention as advanced structural materials because of their heat-resistance properties. In this paper, a study on the vibration of cylindrical shells made of a functionally gradient material (FGM) composed of stainless steel and nickel is presented. The objective is to study the natural frequencies, the influence of constituent volume fractions and the effects of configurations of the constituent materials on the frequencies. The properties are graded in the thickness direction according to a volume fraction power-law distribution. The results show that the frequency characteristics are similar to that observed for homogeneous isotropic cylindrical shells and the frequencies are affected by the constituent volume fractions and the configurations of the constituent materials. The analysis is carried out with strains–displacement relations from Love’s shell theory and the eigenvalue governing equation is obtained using Rayleigh–Ritz method. The present analysis is validated by comparing results with those in the literature.     

Containment of explosions in cylindrical shells

An experimental and theoretical investigation of the high explosive containment ability of circular cylindrical shells is presented. Approximate expressions for final circumferential strain or radial displacement as a function of cylinder length are developed in terms of elementary functions, based upon an assumed rigid-plastic material relation for the containment vessel. The material relation includes strain hardening and approximate strain-rate sensitivity. The expressions are presented in a form convenient for containment design purposes and are shown to be in reasonable agreement with experimental results for several container materials and radius-to-thickness ratios.     

Nonlinear dynamic response of rotating circular cylindrical shells with precession of vibrating shape—Part II: Approximate analytical solution

In this paper the method of harmonic balance is applied to study the nonlinear dynamic response in forced oscillations of a third-order nonlinear partial differential system obtained in Part I of this study. By using improved mode expansion examined in Part I, attention is concerned on the dynamic response of a rotating circular cylindrical shell with respect to the effect of precession of vibrating shape in the spectral neighborhood of one of the lowest natural frequencies. It can be found that the results obtained with the method developed in this study agree with numerical simulation in Part I very well, which indicate that this method has a good accuracy and is efficient for the dynamic analysis of rotating circular cylindrical shells. The stability of the period solutions is also examined in detail.     

A study on stress analysis of orthotropic composite cylindrical shells with a circular or an elliptical cutout

The stress analysis on orthotropic composite cylindrical shells with one circular or one elliptical cutout subjected to an axial force is carried out by using an analytical and experimental method. The composite cylindrical shell governing equation of the Donnell's type is applied to this study and all results are presented by the stress concentration factor. The stress concentra-tion factor is defined as the ratio of the stress on the region around a cutout to the nominal stress of the shell. The stress concentration factor is classified into the circumferential stress concen-tration factors and the radial stress concentration factors due to the cylindrical coordinate of which the origin is the center of a cutout. The considered loading condition is only axial tension loading condition. In this study, thus, the maximum stress is induced on perpendicular region against axial direction, on the coordinate. Various cutout sizes are expressed using the radius ratio,, which is the radius of a cutout over one of the cylindrical shell. Experimental results are obtained using strain gages, which are attached around a cutout of the cylindrical shell. As the result from this study, the stress concentration around a cutout can be predicted by using the analytical method for an orthotropic composite cylindrical shell having a circular or an elliptical cutout.     

A study of the plastic buckling of axially compressed cylindrical shells with a thick-shell theory

A thick shell theory is used to calculate the critical load of plastic buckling of axially compressed cylindrical shells. The buckling equations are derived with the principle of virtual work on the basis of a transverse shear deformable displacement field. The deformation theory of plasticity is used for constitutive equations. To fit the uniaxial stress–strain curve, the Ramberg–Osgood equation is used. In the numerical examples special attention is paid to the dependence of the buckling mode on the ratios of radius to thickness R/h and length to radius L/R. This dependence divides the (R/h,L/R)-plane into simply connected regions each of which corresponds to a buckling mode. These regions form a “buckling mode map”.     

Vibration of completely free composite plates and cylindrical shell panels by a higher-order theory

This paper deals with the free vibration of open, laminated composite, circular cylindrical panels having a rectangular plan-form and all their edges free of external tractions. The material arrangement of the shell panels considered may vary from this of the single isotropic (or special orthotropic) layer to that of a general angle-ply lay-up. The analysis is based on the application of the Ritz approach on the energy functional of the Love-type version of a unified shear deformable shell theory. A through-thickness parabolic distribution of the transverse shear deformation is mainly assumed but, for comparison purposes, numerical results that are based on the assumptions of the classical Love-type shell theory are also presented. The Ritz method is a powerful analytical technique since, provided that a complete set of trial functions is employed, it can provide the exact solution of the problem considered in infinite series forms. The mathematical formulation is therefore presented in a general form, appropriate for any set of basis functions. The variational approach is, however, finally applied in conjunction with a complete functional basis made of the appropriate admissible orthonormal polynomials.     

Buckling analysis of discretely stringer-stiffened cylindrical shells

Engineering approach for computation of stringer stiffened cylindrical shells is realized mainly using the structurally orthotropic theory with momentless pre-buckling state. On the other hand, experimental results suggest that in many cases the mentioned theory provides excessive values of buckling load. The influence of imperfections for stringer stiffened shells seems to be less important than in an isotropic case. Considering axially symmetric momentous components of pre-buckling state cannot essentially improve theoretical results. Specific experiments showed a significant influence of stringer discreteness on the buckling loads of reinforced shells. The mentioned influence can be divided into two parts: excitation of essentially non-axially symmetric pre-buckling and buckling states. Usually, only the latter phenomenon is taken into account. In this paper we show that the first factor dominates. We propose simple analytical expressions governing non-axially symmetric pre-buckling state components. We also propose an asymptotic simplification of the buckling boundary value problem. Results obtained are compared numerically with the known theoretical and experimental data.     

Vibration of cylindrical shells with ring support

In this paper, a study on the vibration of thin cylindrical shells with ring supports is presented. The cylindrical shells have ring supports which are arbitrarily placed along the shell and which imposed a zero lateral deflection. The study is carried out using Sanders' shell theory. The governing equations are obtained using an energy functional with the Ritz method. Results are presented on the frequency characteristics, influence of ring support position and the influence of boundary conditions. The present analysis is validated by comparing results with those available in the literature.