Buckling Of Shells Research Articles

Buckling of pressurized spherical shells containing a through-thickness defect

We present a study on the pressure buckling of thin, elastic spherical shells containing a thickness defect. Methodologically, we combine precision model experiments, finite element simulations, and a reduced axisymmetric shell model. We observe qualitatively different buckling behavior by varying the geometry of the defect: either one buckling event or two events comprising local buckling at the defect and global buckling of the entire shell. We systematically analyze the loading path for the imperfect shell under prescribed pressure or volume change and identify three buckling regimes. We then explore a wide parameter space to study the dependence of the buckling regimes on the defect geometry, thus obtaining a phase diagram with quantitative relationships between critical buckling pressures and defect geometry. We find that the global buckling becomes insensitive to the defect beyond a critical value of its amplitude, and we demonstrate that the buckling regimes are governed by the three geometric parameters of the defect, namely its width, amplitude and the width of the transition region across the edge of the defect.

Nonlinear buckling behavior of functionally graded material sandwich cylindrical shells with tangentially restrained edges subjected to external pressure and thermal loadings

An analytical investigation on buckling and postbuckling behavior of sandwich cylindrical shells comprising functionally graded material layers and subjected to uniform temperature rise and external pressure in thermal environments is presented in this paper. Two sandwich models corresponding to functionally graded material face sheets and core layer are considered. The properties of constitutive materials are assumed to be temperature dependent and effective properties of functionally graded material are determined according to linear rule of mixture. Formulations are based on the classical shell theory taking Von Karman–Donnell nonlinearity and elastic constraint at edges into consideration. Multi-term solutions of deflection and stress function are assumed to satisfy simply supported boundary conditions and Galerkin method is applied to derive nonlinear load–deflection relations from which buckling loads and postbuckling paths are determined. The effects of sandwich configurations, material gradation, tangential constraints of edges, and temperature dependence of material properties on buckling loads and postbuckling curves are analyzed through numerical examples.

An efficient approach for prediction of the nonlocal critical buckling load of double-walled carbon nanotubes using the nonlocal Donnell shell theory

This paper aims to apply the nonlocal Donnell shell theory to study the buckling of double-walled carbon nanotubes (DWCNTs) under axial compression taking into account the effects of internal small length scale and the Van der Waals interactions between layers. The DWCNTs is modeled as nonlocal double circular cylindrical elastic shells. On the basis of nonlocal elasticity theory, governing equations for buckling of Donnell shells are obtained taking into account the Van der Waals force. The nonlocal buckling load of DWCNT is derived without any assumption on radius tubes. However, it is a very difficult task to obtain the analytical solution of nonlocal critical buckling load. In this paper, we develop an approach for prediction of the nonlocal critical buckling load of DWCNTs.

Confined elasticae and the buckling of cylindrical shells

Abstract For curves of prescribed length embedded into the unit disk in two dimensions, we obtain scaling results for the minimal elastic energy as the length just exceeds 2 ⁢ π {2\pi} and in the large length limit. In the small excess length case, we prove convergence to a fourth-order obstacle-type problem with integral constraint on the real line which we then solve. From the solution, we obtain the energy expansion 2 ⁢ π + Θ ⁢ δ 1 3 + o ⁢ ( δ 1 3 ) {2\pi+\Theta\delta^{\frac{1}{3}}+o(\delta^{\frac{1}{3}})} when a curve has length 2 ⁢ π + δ {2\pi+\delta} and determine first order coefficient Θ ≈ 37 {\Theta\approx 37} . We present an application of the scaling result to buckling in two-layer cylindrical shells where we can determine an explicit bifurcation point between compression and buckling in terms of universal constants and material parameters scaling with the thickness of the inner shell.

Cylindrical shells under uniform bending in the framework of Reference Resistance Design

The resistance of cylindrical shells and tubes under uniform bending has received significant research attention in recent times, with a number of major projects aiming to characterise their strength through both experimental and numerical studies. However, the investigated cross-section slenderness ranges have mostly addressed low radius to thickness ratios where buckling occurs after significant plasticity and the influence of geometric imperfections is relatively minor. The behaviour under uniform bending of thinner imperfection-sensitive cylinders that fail by elastic buckling was largely omitted, as was the influence of finite length effects. The value of such resistance models that are only useful for thicker cylinders is therefore somewhat limited.This paper offers the most comprehensive known characterisation of the buckling and collapse resistance of isotropic cylindrical shells and tubes under uniform bending. Expressed within the modern framework of Reference Resistance Design (RRD), it holistically incorporates the effects of material plasticity, geometric nonlinearity and sensitivity to realistic and damaging weld depression imperfections. The characterisation was made possible by the authors' recently-developed novel methodology for mass automation of nonlinear shell buckling finite element analyses. A modification of the RRD formulation is proposed which facilitates its application to systems of low slenderness, and offers a compact algebraic characterisation of all potential imperfection amplitudes for this common shell structural condition. A reliability analysis is also performed.

Elastic local shell and stiffener-tripping buckling strength of ring-stiffened cylindrical shells under external pressure

Abstract Ring-stiffened cylindrical shells are often used in offshore and submergible structures, it is important for the structural safety to estimate the buckling strength of those under extremely high water pressure. Although the buckling strength can be accurately calculated using the finite element analysis (FEA), the estimation by more easy way is required at an initial stage of the structural design. This paper proposes two formulas for the local shell and stiffener-tripping buckling strength of the ring-stiffened cylindrical shells under external pressure. One can estimate the tripping buckling strength only and is derived assuming the tripping buckling as buckling of only the flange-beam supported by springs and considering effects of the cylindrical shape and torsional stiffness which are not included in a conventional formula. The other can estimate both the shell and tripping buckling strength and is considered interaction of buckling deflection between a cylindrical shell and ring-stiffeners and influence of stresses acting on the web which are not included in conventional formulas. Based on the principle of minimum potential energy, the formula is derived using the functions to express the buckling deformation in the cylindrical shell and ring-stiffeners. The buckling strength estimated by the two proposed formulas is compared with that by existing conventional formulas and the finite element analysis, and this study discusses the influence of new considerations on the shell and tripping buckling strength. From the results, it is found that the second formula can deal with any buckling mode and has greatly high accuracy compared with other conventional formulas.

Postbuckling analysis of hydrostatic pressurized FGM microsized shells including strain gradient and stress-driven nonlocal effects

Herein, with the aid of the newly proposed theory of nonlocal strain gradient elasticity, the size-dependent nonlinear buckling and postbuckling behavior of microsized shells made of functionally graded material (FGM) and subjected to hydrostatic pressure is examined. As a consequence, the both nonlocality and strain gradient micro-size dependency are incorporated to an exponential shear deformation shell theory to construct a more comprehensive size-dependent shell model with a refined distribution of shear deformation. The Mori–Tanaka homogenization scheme is utilized to estimate the effective material properties of FGM nanoshells. After deduction of the non-classical governing differential equations via boundary layer theory of shell buckling, a perturbation-based solving process is employed to extract explicit expressions for nonlocal strain gradient stability paths of hydrostatic pressurized FGM microsized shells. It is observed that the nonlocality size effect causes to decrease the critical hydrostatic pressure and associated end-shortening of microsized shells, while the strain gradient size dependency leads to increase them. In addition, it is found that the influence of the internal strain gradient length scale parameter on the nonlinear instability characteristics of hydrostatic pressurized FGM microsized shells is a bit more than that of the nonlocal one.

Elastoplastic buckling of a cylindrical shell with initial geometric imperfections and an elastic filler at external pressure

A technique has been developed for the numerical calculation of deformation and elastoplastic stability loss nonlinear problems rotating shells based on the Tymoshenko hypotheses for nonshallow shells taking into account geometric nonlinearities. Kinematic relations are formulated in velocities and constructed in the current status metric taking into account large deformations, displacements and rotation angles of shell elements. plastic flow theory with nonlinear isotropic hardening describes physical relations. The motion equations result from the virtual working power balance. Winkler foundation models the revolution shell and the elastic filler contact interaction. The numerical calculation is based on an explicit “cross” type mesh scheme. The dependence of the elastoplastic revolution shell stability loss form and the load critical value on the filler stiffness under various values of the initial imperfections amplitude was studied.

Исследование прочности и устойчивости ортотропных конических оболочек и конических панелей

Исследование прочности и устойчивости ортотропных конических оболочек и конических панелей

The influence of axial load on elastic buckling of shells of revolution

The influence of axial load on elastic buckling of shells of revolution

Effects of initial form-finding models on shape, force paths, and buckling of anti-funicular shells with free edges

This paper presents an analysis on the effect of the initial form-finding models properties on the shape of shells with free edges, and consequently on the structural behaviour concerning force paths and buckling. The stiffness distribution of these initial models determines the membrane forces that equilibrate the applied loadings and, in this way, the obtained shape. With physical models, this question corresponds to the choice of material: latex rubber membrane or orthotropic fabrics. Continuous models with isotropic materials lead to a higher distribution of forces and reduced curvatures, without concentration along the free edges; this is in contrast to what occurs in discrete models without in-plane shear stiffness. In shells with free edges, the effects on the structural behaviour are related to curvatures and even more dependent on the initial model properties. Solutions with higher curvatures have higher concentrations of forces in the free edges. The reduction of curvatures and forces along the free edges of shells with an approximately square plan is favourable; this is in contrast to shells with elongated rectangular or triangular plans. The presented results and discussion aim to contribute to the design of new thin-shell structures.

Generalized variational principles for buckling analysis of circular cylinders

A generalized variational principle and a parameterized generalized variational principle are obtained for large deformation analysis of circular cylinders by the semi-inverse method; all known variational principles in the literature are special cases of the obtained parameterized functional. In this approach, a trial functional is constructed with an energy-like integral involving an unknown function, which is identified step by step. The present paper provides a quite straightforward but rigorous tool to the construction of a variational principle for the shell or plate buckling.

Nonlinear buckling of orthogonal carbon nanotube-reinforced composite cylindrical shells under axial compression surrounded by elastic foundation in thermal environment

Nonlinear buckling and postbuckling of orthogonal carbon nanotube-reinforced composite (Orthogonal CNTRC) cylindrical shells subjected to axial compression in thermal environments surrounded by elastic foundation are presented in this paper. Two layers of shell are reinforced by carbon nanotube (CNT) in two orthogonal directions (longitudinal and circumferential directions). Based on Donnell’s shell theory with von Karman’s nonlinearity and the Galerkin method, the governing equations are established to obtain the critical buckling loads and postbuckling load-deflection curves. The large effects of CNT volume fraction, temperature change, elastic foundation and geometrical parameters of cylindrical shells on the buckling load and postbuckling behavior of Orthogonal CNTRC cylindrical shells are obtained.

Harnessing energy landscape exploration to control the buckling of cylindrical shells

Even for relatively simple thin shell morphologies, many different buckled configurations can be stable simultaneously. Which state is observed in practice is highly sensitive to both environmental perturbations and shell imperfections. The complexity and unpredictability of postbuckling responses has therefore raised great challenges to emerging technologies exploiting buckling transitions. Here we show how the buckling landscapes can be explored through a comprehensive survey of the stable states and the transition mechanisms between them, which we demonstrate for cylindrical shells. This is achieved by combining a simple and versatile triangulated lattice model with efficient high-dimensional free-energy minimisation and transition path finding algorithms. We then introduce the method of landscape biasing to show how the landscapes can be exploited to exert control over the postbuckling response, and develop structures which are resistant to lateral perturbations. These methods now offer the potential for studying complex buckling phenomena on a range of elastic shells.

Experimental and numerical investigations of the ultimate torsional strength of an ultra large container ship

Structures of ultra large container ships (ULCS) are characterized by large deck openings and low torsional rigidity. It is essential to comprehensively figure out their collapse behaviors under pure torsion with both model experiments and numerical simulations, making an evaluation of their ultimate torsional strength. In this paper, a similar scale model of a 10,000TEU container ship has been designed and manufactured first, in which both geometric similarity and strength similarity are taken into account. Next the collapse behaviors of the test model are detailedly illustrated with both experimentally and numerically obtained results. Then discussions on warping or shear buckling deformations involved in the collapse process of the structure are conducted with extended numerical simulations. Finally, the ultimate torsional strength of the true ship is evaluated according to the similarity theory. Results show that it is the yielding and shear buckling of the side shells that causes the failure of the hull girder under pure torsion. Further nonlinear finite element analysis demonstrates that it may either have warping or shear buckling deformations in the torsional collapse process of the hull girder with a large deck opening, depending on the local rigidity distribution of side shells, which has a significant effect on the ultimate torsional strength of the hull girder.

Numerical Analysis of Sandwich Composite Deep Submarine Pressure Hull Considering Failure Criteria

The pressure hull is the primary element of submarine, which withstands diving pressure and provides essential capacity for electronic systems and buoyancy. This study presents a numerical analysis and design optimization of sandwich composite deep submarine pressure hull using finite element modeling technique. This study aims to minimize buoyancy factor and maximize deck area and buckling strength factors. The collapse depth is taken as a base in the pressure hull design. The pressure hull has been analyzed using two composite materials, T700/Epoxy and B(4)5505/Epoxy, to form the upper and lower faces of the sandwich composite deep submarine pressure hull. The laminated control surface is optimized for the first ply failure index (FI) considering both Tsai–Wu and maximum stress failure criteria. The results obtained emphasize an important fact that the presence of core layer in sandwich composite pressure hull is not always more efficient. The use of sandwich in the design of composite deep submarine pressure hull at extreme depths is not a safe option. Additionally, the core thickness plays a minor role in the design of composite deep submarine pressure hull. The outcome of an optimization at extreme depths illustrates that the upper and lower faces become thicker and the core thickness becomes thinner. However, at shallow-to-moderate depths, it is recommended to use sandwich composite with a thick core to resist the shell buckling of composite submarine pressure hull.

Optimal Control of SonoVue Microbubbles to Estimate Hydrostatic Pressure.

The measurement of cardiac and aortic pressures enables diagnostic insight into cardiac contractility and stiffness. However, these pressures are currently assessed invasively using pressure catheters. It may be possible to estimate these pressures less invasively by applying microbubble ultrasound contrast agents as pressure sensors. The aim of this study was to investigate the subharmonic response of the microbubble ultrasound contrast agent SonoVue (Bracco Spa, Milan, Italy) at physiological pressures using a static pressure phantom. A commercially available cell culture cassette with Luer connections was used as a static pressure chamber. SonoVue was added to the phantom, and radio frequency data were recorded on the ULtrasound Advanced Open Platform (ULA-OP). The mean subharmonic amplitude over a 40% bandwidth was extracted at 0–200-mmHg hydrostatic pressures, across 1.7–7.0-MHz transmit frequencies and 3.5%–100% maximum scanner acoustic output. The Rayleigh–Plesset equation for single-bubble oscillations and additional hysteresis experiments were used to provide insight into the mechanisms underlying the subharmonic pressure response of SonoVue. The subharmonic amplitude of SonoVue increased with hydrostatic pressure up to 50 mmHg across all transmit frequencies and decreased thereafter. A decreasing microbubble surface tension may drive the initial increase in the subharmonic amplitude of SonoVue with hydrostatic pressure, while shell buckling and microbubble destruction may contribute to the subsequent decrease above 125-mmHg pressure. In conclusion, a practical operating regime that may be applied to estimate cardiac and aortic blood pressures from the subharmonic signal of SonoVue has been identified.

Finite element simulation of Goldmann tonometry after refractive surgery

BackgroundThe problem of accuracy of intraocular pressure (IOP) measurement using Goldmann Applanation Tonometry (GAT) after a refractive surgery is still unresolved.In this study a numerical simulation of refractive surgery and GAT is proposed as a method to evaluate accuracy of the correction formula for GAT reading after changes in corneal curvature and thickness. MethodsIn the finite element analysis the effect of the corneal shell buckling was taken into account. This effect is responsible for the nonlinear relation of Goldmann applanation tonometer measurement and intraocular pressure. Goldmann applanation tonometry was simulated for different ablation depths and values of intraocular pressure levels. FindingsThe results of the simulation indicate a strong relation between the Goldmann applanation reading and the intraocular pressure level for values greater than nominal (about 16 mmHg). The estimation error of the intraocular pressure after changes in corneal geometry, is lower than 1 mmHg for IOP = 32 mmHg and lower than 3 mmHg for IOP = 48 mmHg. InterpretationThe proposed correction formula of Goldmann applanation tonometry occurred to be useful in the cases after the corneal refractive surgery regardless of the intraocular pressure values and ablation depth.

A Weak Form Implementation of Nonlinear Axisymmetric Shell Equations With Examples

Abstract We present a weak form implementation of the nonlinear axisymmetric shell equations. This implementation is suitable to study the nonlinear deformations of axisymmetric shells, with the capability of considering a general mid-surface shape, non-homogeneous (axisymmetric) mechanical properties and thickness variations. Moreover, given that the weak balance equations are arrived to naturally, any external load that can be expressed in terms of an energy potential can, therefore, be easily included and modeled. We validate our approach with existing results from the literature, in a variety of settings, including buckling of imperfect spherical shells, indentation of spherical and ellipsoidal shells, and geometry-induced rigidity (GIR) of pressurized ellipsoidal shells. Whereas the fundamental basis of our approach is classic and well established, from a methodological view point, we hope that this brief note will be of both technical and pedagogical value to the growing and dynamic community that is revisiting these canonical but still challenging class of problems in shell mechanics.

Comparison accurate calculate and Numerical solution of buckling thin Cylindrical shape memory alloy shell under Uniform load with d.q.m

The shape memory alloy has found too much significance among engineers and researchers due to their mechanical features behaviours and these alloys features are changed by temperature and tension. Also in correspondence to great mechanical strain made due to none linear and geometric features emerged in structure, encounter the equations solution with difficulty. In this research the numerical number has been presented based on the DQM basis to axisymmetric buckling of cylindrical shells made of shape memory alloys. First the solution incorporation presented then the numerical results are compared with exact solutions