Thin-walled Circular Cylindrical Shells Research Articles

Critical circumferential wavelength of elastic buckling of longitudinal compressed thin-walled cylindrical shells

The classical theory of elastic critical buckling stress works well for slender columns and thin flat plates under compression; however, the situation is different for longitudinally compressed thin-walled circular cylindrical shells, and the issue has plagued us despite considerable efforts over the last 100 years. We noticed that all such buckling analyses thus far, both linear and nonlinear, in terms of the main philosophy, inherited and were confined to Euler’s pioneering solution for the slender column model that focuses on the longitudinal buckling deformation mode and should be classified as the ‘longitudinal open-loop’ eigenmode because the deformations of the two longitudinal ends are physically independent of each other. In view of this, for an ideal linear-elastic buckling model of a thin-walled perfectly circular cylindrical shell under uniform longitudinal compression on the foundation of the longitudinal open-loop eigenmode solution, it is also necessary to consider a ‘circumferential closed-loop’ eigenmode simultaneously to physically avoid violating the reality of its ideal periodic deformation on the entire perimeter and to mathematically redefine the biunique and precise relationship for each distinct eigenmode by the critical circumferential wavelength. Originating from such a case study, the mathematical uniqueness issue hidden in the general solution of the Donnell equation is further discussed. The authenticity of the competing eigenmode characterized by the Koiter circle is also discussed. Furthermore, a preliminary attempt was conducted to interpret the discrepancy between theoretical and experimental buckling loads, mainly initiated by the characteristic type of longitudinally generated circumferential local inward displacement in initial geometric imperfections, using the insights herein.

Testing and Analysis of Additively Manufactured Stainless Steel Corrugated Cylindrical Shells in Compression

Initial geometric imperfections have been identified as the main cause for the large discrepancies between experimental and theoretical buckling loads of thin-walled circular cylindrical shells under axial compression. The extreme sensitivity to imperfections has been previously addressed and mitigated through the introduction of stiffeners; however, sensitivity still remains. Optimized corrugated cylindrical shells are largely insensitive to imperfections and hence exhibit excellent load-bearing capacities, but their complex geometries make their construction difficult and costly using conventional manufacturing techniques. This was overcome in the present study through additive manufacturing (AM). Nine optimized corrugated shells with different diameter-to-thickness ratios, together with one reference circular cylindrical shell, were additively manufactured by means of powder bed fusion (PBF) from austenitic and martensitic precipitation hardening stainless steel. The structural behavior of the AM shells was then investigated experimentally with the testing program comprising tensile coupon tests, measurements of basic geometric properties, and axial compression tests. Numerical analyses were also conducted following completion of the physical experiments. The experimental and numerical results verified the effectiveness of optimized corrugated cylindrical shells in achieving improved local buckling capacity and reduced imperfection sensitivity. Initial recommendations for the structural design of the studied cross-sections are made.

On the stability of thin-walled circular cylindrical shells under static and periodic radial loading

In this paper, we study the dynamic behavior of the cylindrical shells subjected to static and time-varying (periodic) radial loading. Flügge-Luré-Byrne shell theory has been used to derive dynamic equations in the form of nonlinear partial differential equations (PDEs). Linearization of PDEs about the steady-state periodic solutions, followed by Galerkin projection, yields a system of ordinary differential equations (ODEs) with time-varying periodic coefficients. Those are similar to the higher-order Mathieu-Hill equations and their stability analyses have been ascertained using Floquet theory. Dynamic instabilities of the cylindrical shells have been presented through stability charts in the plane of forcing parameters, namely, the static component of the pressure, the amplitude, and frequency of the fluctuating component of the pressure. Also, possible sub and super harmonic resonances have been explored through the stability charts. Four different boundary conditions, viz., simply supported, clamped–free, pinned–free (axial constraint at the pinned end) and pinned–free (axial constraint at the free end) have been considered in the present analysis. Presented charts can be very useful during the design of the shell structures subjected to static/dynamic loading.

Natural Frequencies Optimization of Thin-Walled Circular Cylindrical Shells Using Axially Functionally Graded Materials.

One method to avoid vibration resonance is shifting natural frequencies far away from excitation frequencies. This study investigates optimizing the natural frequencies of circular cylindrical shells using axially functionally graded materials. The constituents of functionally graded materials (FGMs) vary continuously in the longitudinal direction based on a trigonometric law or using interpolation of volume fractions at control points. The spatial change of material properties alters structural stiffness and mass, which then affects the structure’s natural frequencies. The local material properties at any place in the structure are obtained using Voigt model. First-order shear deformation theory and finite element method are used for estimating natural frequencies, and a genetic algorithm is used for optimizing material volume fractions. To demonstrate the proposed method, two optimization problems are presented. The goal of the first one is to maximize the fundamental frequency of an FGM cylindrical shell by optimizing the material volume fractions. In the second problem, we attempt to find the optimal material distribution that maximizes the distance between two adjoining natural frequencies. The optimization examples show that building cylindrical shells using axially FGM is a useful technique for optimizing their natural frequencies.

SN-EVM unit for experimental studies of stability in circular cylindrical shells under combined loading

The article summarizes results of experimental studies of stability in circular cylindrical shells under combined loading by compressive force, internal pressure and torsion at the SN-EVM unit, developed by the Department for Strength of Materials, Theory of Elasticity and Plasticity at Tver State Technical University. It gives description of the test facility and methodology of experiments for studying the elastoplastic properties of materials under combined loading. It cites the processed experimental results for comparison with the formulated theory of bifurcation for the thin-walled circular cylindrical shell.

Buckling and post-buckling behavior of various dented cylindrical shells using CFRP strips subjected to uniform external pressure: Comparison of theoretical and experimental data

Thin-walled circular cylindrical shells are used in many civil engineering applications. Any initial geometric imperfections and the type of applied load determine the buckling and the post buckling of cylindrical thin-walled shell structures. Fourteen thin-walled cylindrical shell models in two groups (labelled without CFRP and with CFRP) with different dent directions (H= horizontal, V= vertical and D= 45° diagonal direction) and dent depths (2tc and 4tc) subjected to external pressure were tested in the present research (tc is the thickness of the cylindrical shell). The results of testing under different theories and codes are compared. Moreover, the results are identified and investigated in the present experimental research.

Deformation of cylindrical shells of steel 45 under complex loading

Reviewing the numerical simulation of the process of non-proportional elastic-plastic deformation of steel 45 by a knot of constant curvature, taking into account the complex nature of deformation under complex subcritical loading by axial compressive force and torque for a thin-walled circular cylindrical shell. The theory of Quas and simple processes of A. A. Ilyushin and the mathematical model of V. Zubchaninov were applied taking into consider the parameters of the complex loading for plane trajectories To assess the accuracy of accepted theories, the simulation results are compared with experimental results, received on the automated complex СNcomputer in the laboratory of the faculty of «Strength of materials and theory of elasticity and plasticity» of the Tver state technical University. Was introduced the scheme of calculations disproportionate plastic deformation of steel 45 using the proposed mathematical model showed a satisfactory result and recommended for further use. Remarks, that in the described processes the lack of some parameters complex loading in approximations reduces the accuracy of the final calculated values, differences significantly compared to the experimental data.

Artificial neural network prediction of buckling load of thin cylindrical shells under axial compression

Thin-walled circular cylindrical shells under axial compression are prone to buckling; the reduction of buckling load from the theoretical estimation is considered primarily due to imperfection sensitivity. The buckling load from carefully conducted experiments using nominally similar shells falls below the prediction by the classical theory with substantial scatter. The current design recommendations apply highly conservative knockdown factors to the theoretical buckling loads to estimate the load carrying capacity of the shell structures. In this study, a systematic analysis of experimental data from the literature has been conducted using the artificial neural network (ANN). The networks were trained using Bayesian regularisation backpropagation training function. Two network models with eight and ten neurones were used to train, test and validate 390 sets of experimental data. The buckling loads predicted by the ANN models were compared with the design recommendations by National Aeronautics and Space Administration (NASA), Eurocode 3 (EC3) and the experimental buckling loads. The ANN models predict buckling load within 10% of the experimental buckling load and can be reliably used within the parametric range used in training. The NASA design recommendations provides 10–50% conservative estimates compared to the experimental loads while EC3 predictions are conservative by more than 50%.

Effect of radial loads on the natural frequencies of thin-walled circular cylindrical shells

In this paper, we study the variation of natural frequencies for thin walled circular cylindrical shells subject to external radial load (uniform and hydrostatic pressure). The study has been performed for four different types of boundary conditions viz. pinned-pinned, pinned-free, clamped-free and one end is pinned and other end axially constrained. Towards this, we have used approximate shape functions and Galerkin projections of the PDEs governing the shell motion, linearized about a steady state solution due to the applied mean radial pressure. Our results have very good agreement with the experimental and theoretical results available in literature. They have also been verified against finite element analysis (FEA) using ABAQUS. We have found that with increasing magnitude of inward radial load, the natural frequency decreases, however, the circumferential wavenumber corresponding to the lowest natural frequency increases. For outward radial load the opposite happens. Furthermore, the circumferential wavenumbers corresponding to lowest natural frequency and first buckling mode are different for short cylinders with the difference decreasing with increasing length of the cylinder. We have also found an interesting result for the beam mode vibration of cylindrical shells. For short cylinders, the natural frequency of the beam mode increases (decreases) with internal (external) pressure, which is consistent with other circumferential modes. Subsequently, we obtain a critical length at which the applied radial pressure has no effect on the natural frequency. For cylinders longer than the critical length, the effect of the radial load is opposite of that observed for the short cylinders, and this is contrary to the behavior of the other circumferential modes. As a result, buckling of the beam mode happens for an outward radial (internal) pressure.

Natural Vibrations and Stability of Elliptical Cylindrical Shells Containing Fluid

The paper deals with a three-dimensional problem on natural vibrations and stability of thin-walled cylindrical shells with arbitrary cross sections, containing a quiescent or flowing ideal compressible fluid. The motion of compressible non-viscous fluid is described by a wave equation, which together with the impermeability condition and corresponding boundary conditions is transformed using the Bubnov–Galerkin method. A mathematical formulation of the problem of thin-walled structure dynamics has been developed based on the variational principle of virtual displacements. Simulation of shells with arbitrary cross sections is performed under the assumption that a curvilinear surface is approximated to sufficient accuracy by a set of plane rectangular elements. The strains are calculated using the relations of the theory of thin shells based on the Kirchhoff–Love hypothesis. The developed finite element algorithm has been employed to investigate the influence of the fluid level, the ratio of the ellipse semi-axes and types of boundary conditions on the eigenfrequencies, vibration modes and the boundary of hydroelastic stability of thin-walled circular and elliptical cylindrical shells interacting with a quiescent or flowing fluid.

Buckling Analysis of Nonlocal Anisotropic Thin-Walled Cylindrical Shells Subject to Combined Loading

AbstractThe equilibrium governing equations of nonlocal anisotropic thin-walled circular cylindrical shell under combined axial compressive force, torsional load, and external pressure are explicitly derived. This is accomplished by appropriately combining the equilibrium equations and the strain-displacement relations according to Flugge’s shell theory and the stress-stain equations of Eringen’s nonlocal elasticity theory. An analytical solution for the buckling of the shells is presented by using the complex method. This model is validated by a good agreement between the results given by the present model and available data in the literature. Furthermore, the model is utilized to elucidate the buckling properties for different load combinations.

Instabilities of thin circular cylindrical shells under radial loading

We study the stability of thin walled circular cylindrical shells subject to radial pressure loading. Towards this end, we first develop the equations of motion of these shells with predominant radial deflections. We identify additional terms in the shell equations of motion which have mostly been neglected in existing studies and ascertain their importance in predicting the correct static buckling pressure. For finite cylinders, incorporation of these terms results in the best prediction of the static buckling pressure. Furthermore, we also find that the assumed relationships between the in-plane displacements of a generic point and the in-plane displacements of the mid-surface, which change from one shell theory to the other, play a critical role in determining the correct onset of the buckling instability under radial loading. In particular, the Donnell shell theory which is very popular for the study of dynamic problems related to cylindrical shells does not predict the correct buckling pressure. We find that the Flügge–Luré–Byrne theory is best suited for this purpose. Accordingly, the shell equations with the additional terms and the in-plane displacements related as per the Flügge–Luré–Byrne shell theory have been used for the study of parametric instability of a cylinder subject to a uniform radially fluctuating pressure. Stability charts with respect to different combinations of the forcing parameters viz. the static component of the pressure, the amplitude and frequency of the fluctuating component of the pressure have been presented which can serve as design guideline for shells subject to fluctuating radial loads.

Stability of elastoplastic thin-walled structures in implementation of complex processes of combined deformation

In this article the problem of bifurcation of the thin-walled circular cylindrical shell under compound deformation in the moment of loss of stability is being considered. Undercritical loading is compound. It is full failed by means of axial compressing force and torque moment in the deviatoric plane A.A. Ilyushin. The condition of the incompressibility of material and the condition of the uniformity of the stressed state in the shell till the moment of loss of stability are used. The problem is solved in geometrically linear setting.

Comparison of theoretical approaches to account for geometrical imperfections of unstiffened isotropic thin walled cylindrical shell structures under axial compression

Thin walled circular cylindrical shell structures are prone to buckle and very sensitive towards geometrical imperfections. The influence of imperfections on the load carrying capacity of shell structures, as they are applied in launcher vehicles is considered by reducing theoretical buckling loads with empirical knock down factors. In general, these knock down factors may lead to a conservative estimate of the load carrying capacity since a worst case scenario is considered. In order to exploit the lightweight design potential of a structure, theoretical approaches to account for geometrical imperfections may lead to more adequate buckling load predictions. Within this contribution different theoretical approaches to account for geometrical imperfections of isotropic shell structures subjected to axial compression are investigated, and the influence of these approaches on the buckling load obtained is studied.

Study on the optimization of cutting parameters in turning thin-walled circular cylindrical shell based upon cutting stability

Thin-walled circular cylindrical shell, one of the most typical difficult-to-cut workpieces, is prone to vibration and regenerative vibration especially during the cutting process. Being different from previous studies, this paper established a model of cutting dynamics that accounts for both workpiece deformation and tool vibration. The mathematical model of optimization on cutting parameters was built, with constraint conditions and objective function including cutting force, cutting power, surface roughness, cutting stability, and cutting efficiency. The optimal cutting parameters were subsequently acquired by adding the critical cutting width to the mathematical model. Finally, the orthogonal table of the three factors and five levels was adopted in accordance with the orthogonal experimental design, and the material removal rate improves about 8 % by implementing the program of particle swarm optimization. The dynamic tests of cutting process were conducted, verifying the stability of the optimal results that ensures the best surface quality.

Young’s Modulus Identification by Using Cylindrical Specimens

A method for determining the flexural Young’s modulus of polymeric materials from deformation diagrams of thin-walled circular cylindrical shells in compression in the region of geometrical nonlinearity has been elaborated. A numerical solution is found by the finite-element method (ANSYS.) The existence of a unified deformation diagram in generalized coordinates is established, from which the flexural Young’s modulus is determined. To validate the method, the Young’s modulus of specimens was found experimentally.

Solution of the problem of bifurcation of a cylindrical shell considering the complex nature of deformation at the time of loss of stability under complex loading

The authors consider the problem of bifurcation of a thin-walled circular cylindrical shell taking into account the complex nature of the deformation at the time of loss of stability under complex loading by sub-critical axial compressive force and torque in the deviatoric A. Ilyushin plane of strains .

Identification of the elastic modulus of polymeric materials by using thin-walled cylindrical specimens

A method for determining the elastic modulus of polymeric materials from deformation diagrams of thin-walled circular cylindrical shells in compression and tension in the region of geometrical nonlinearity has been elaborated. A numerical solution is found by the finite-element method (ANSYS.) The existence of a unified deformation diagram in generalized coordinates is established, from which the elastic modulus is determined. To validate the method, the eigenfrequencies of cylindrical specimens were found experimentally. The results obtained are compared with FEM calculations.

Determination of stress–strain characteristics of thin polymer films on cylindrical specimens

An experimental method (TWCS method) was applied to determine the flexural elastic modulus E and some other stress-deformation characteristics of a thin-walled circular cylindrical polymer shell at compression. A methodology of test sample preparation and fixation as well as means of the measurement of the load P and the respective displacement ∆ were developed. Two modes of testing the equipment were used in which the sequences of measuring P ∆ → and P ∆ → were implemented. Repeatability of measurements was evaluated. Optimal geometric parameters of test samples and the range of determinable values of modulus E were defined. A procedure of the calculation of the relationship between the tensile stress σ and relative elastic tensile deformation e in the most deformed area of the sample was developed. Applicability of the TWCS method for multiple assessments of changes of deformational characteristics by using a single sample was proved. It was shown that creep, creep recovery, and stress relaxation tests can be performed by the TWCS method.

Analysis of circular cylindrical shells under harmonic forces

A generalized thin shell theory for the stress-deformation analysis of thin-walled circular cylindrical shells is formulated based on the Hamilton's variational principle. The theory is applicable to multiple in-phase and out-of-phase harmonic forces and general boundary conditions. The stationary conditions of the Hamilton functional are then evoked to recover the equations of motion and boundary conditions of the problem. The unknown displacement fields are expanded as infinite complex Fourier series in the circumferential coordinate. The resulting field equations are observed to couple the radial, tangential and longitudinal displacement contributions within each mode, while uncoupling the contributions of a given Fourier mode from those of other Fourier modes. A general solution for the obtained field equations is developed for harmonic loading of general spatial distribution. The method developed is verified against well established solutions and its applicability to practical problems is illustrated through examples.