Buckling Of Shells Research Articles

Modeling the Effect of Preliminary Quasi-Static Loading on the Dynamic Buckling of Composite Cylindrical Shells under Axial Compressive Loading

Modeling the Effect of Preliminary Quasi-Static Loading on the Dynamic Buckling of Composite Cylindrical Shells under Axial Compressive Loading

Влияние распределенных дислокаций на устойчивость полого нелинейно-упругого шара

The phenomenon of stability loss of a hollow elastic sphere containing distributed dislocations and loaded with external hydrostatic pressure is studied. The study was carried out in the framework of the nonlinear elasticity theory and the continuum theory of continuously distributed dislocations. An exact statement and solution of the stability problem for a three-dimensional elastic body with distributed dislocations are given. The static problem of nonlinear elasticity theory for a body with distributed dislocations is reduced to a system of equations consisting of equilibrium equations, incompatibility equations with a given dislocation density tensor, and constitutive equations of the material. The unperturbed state is caused by external pressure and a spherically symmet-ric distribution of dislocations. For distributed edge dislocations in the framework of a harmonic (semi-linear) mate-rial model, the unperturbed state is defined as an exact spherically symmetric solution to a nonlinear boundary value problem. This solution is valid for any function that characterizes the density of edge dislocations. The perturbed equilibrium state is described by a boundary value problem linearized in the neighborhood of the equilibrium. The analysis of the axisymmetric buckling of the sphere was performed using the bifurcation method. It consists in determining the equilibrium positions of an elastic body, which differ little from the unperturbed state. By solving the linearized problem, the value of the external pressure at which the sphere first loses stability is found. The effect of dislocations on the buckling of thin and thick spherical shells is analyzed.

Coupled nonlinear mechanics for the electromechanical response of multi-stable piezoelectric shallow shells with piezoelectric films

This paper considers the nonlinear electromechanical response of shallow piezoelectric laminated shells of arbitrary double curvature, undergoing to snap through buckling and eventually to elastic instability, under mechanical pressure loads. The computational framework is based on coupled nonlinear mechanics encompassing the electromechanical coupling field and geometric nonlinearity due to large displacements and rotations. The first shear deformation theory is combined with a linear layerwise theory for describing the elastic and electric state variables through the laminate thickness, respectively. The governing equations are expressed with respect to an orthogonal curvilinear coordinate system. Building upon the nonlinear multifield mechanics an eight-node shell finite element is developed to discretize the generalized coupled nonlinear equations, which are subsequently linearized and iteratively solved, using the Cylindrical Arc-Length method in combination with the Newton-Raphson iterative technique. Numerical results demonstrate the capability of the present model to predict the complex electromechanical response of shallow cylindrical shells that transit between two stable equilibrium states. The electromechanical (generated electric charge-energy) conversion is highly complex (depending on both nonlinear piezoelectric and stiffness matrices) and predicted along the stable and unstable paths of bi-stability. At the same time the feasibility to relate the sensory electric potential with the snap through buckling of smart shells is also quantified. The influence of shell thickness, curvature and laminate configuration on the electromechanical conversion, is finally investigated.

원통 복합재 쉘의 비선형 좌굴에 의한 파괴현상 연구

A cylindrical shell is a representative structure that exhibits buckling. To investigate the nonlinear behavior of this shell, the arc-length method or load–displacement control Newton–Raphson numerical analysis method is used. In this study, the buckling and post-buckling phenomena of composite shells were investigated, and the fracture phenomena of composites were evaluated for their load–displacement behaviors. Composite buckling is a complex process compared to those of isotropic materials. Moreover, in the case of a laminated composite material, it is difficult to accurately identify the time and place at which a failure might occur because various stress components and interlayer shear stresses interact with each other. Using the arc-length nonlinear numerical analysis method, the failure phenomena were clearly identified along the behavioral path of the composite shell after buckling, and the differences were confirmed via various composite failure theories.

The stability of geometrically nonlinear plate systems under the action of dynamic loads

Relevance. Single-connected and multi-connected plate systems are widely used in construction, aircraft, shipbuilding, mechanical engineering, instrument making. As a result, the study of the stability of geometrically nonlinear plate systems is an urgent topic. But, despite significant achievements in this area, there are still many unsolved problems. Thus, the requests of the above-mentioned areas of application of thin-walled spatial systems require further study of the issue of static and dynamic stability. The aim of the work - development of a method of the dynamic stability analysis of geometrically nonlinear plate systems such as prismatic shells under the action of dynamic compression loads. Methods. A plate system, which is subject to dynamic compression loads in the longitudinal direction, is considered. Kirchhoff - Love hypotheses are taken into account. The material stress-deformation diagram is linear. The displacement of points in the normal direction to the median plane of the plates is determined in the form of the Vlasov expansion. To derive the basic differential equations of stability, the energy method and the variational Vlasov method are used. The extreme value of the total energy is determined using the Euler - Lagrange equation. As a result, a set of basic nonlinear differential equations for studying the buckling of the plate system under the action of dynamic compression loads is obtained. Results. The developed method is used to stability analysis of a geometrically nonlinear prismatic shell with a closed contour of the cross section, under central compression under the action of dynamic loading. The edges of the shell rest on the diaphragm. The buckling of the prismatic shell in the longitudinal direction along one and two half-waves of a sinusoid is studied. The numerical integration of nonlinear differential equations is performed by the Runge - Kutta method. Based on the calculation results, graphs of the dependence of the relative deflection on the dynamic coefficient are constructed. The influence of the rate of change of compression stress, the initial imperfection of the system, and other parameters on the criteria for the dynamic stability of the plate system is investigated.

Analysis of the fracture toughness parameters at the free edge in layered composites

The problem of deformation and elastoplastic buckling of shells of revolution with a thick-walled elastic core under combined static and dynamic loading is formulated in a two-dimensional planar formulation based on two approaches: full-scale modeling within the framework of continuum mechanics and a simplified formulation based on the hypotheses of the theory of shells of the Timoshenko type and the Winkler foundation. Both approaches allow solving the problems of deformation and stability of non-shallow shells on the basis of Timoshenko's hypotheses, taking into account geometric nonlinearities. The statement from the perspective of continuum mechanics makes it possible to approximate the shell in thickness by a number of layers of finite elements. The constitutive relations are formulated in Lagrange variables using a fixed Cartesian coordinate system as a reference one. Kinematic relations are recorded in the metric of the current state. The elastic-plastic properties of shells are described by the theory of plastic flow with isotropic hardening. The equations of motion follow from the balance of the virtual powers of the work. In the first approach, the contact interaction of a shell and an elastic body is modeled by the conditions of nonpenetration along the normal and free slip along the tangent. The nonpenetration conditions are satisfied only in the active phase of the contact interaction; if the contact is broken, they are replaced by conditions on the free surface. In the second approach, the contact interaction of the elastic core with the shell is modeled by the Winkler foundation. Both approaches allow one to describe the nonlinear subcritical deformation of shells of revolution with an elastic core, to determine the limiting (critical) loads in a wide range of loading rates, taking into account the geometric imperfections of the shape. Using both approaches, a numerical simulation of contact interaction problem of an elastoplastic cylindrical shell with a thick-walled elastic core at a quasi-static uniform external pressure is carried out. The study of the influence of the thickness and initial deflection of the shell, as well as the stiffness and thickness of the core, on the value of the critical pressure and the form of buckling has been carried out. Based on these calculations, a conclusion was made about a wide range of applicability of the Winkler foundation model.

Investigation of the Winkler foundation model applicability for describing the contact interaction of elastoplastic shells with a core under external pressure

The problem of deformation and elastoplastic buckling of shells of revolution with a thick-walled elastic core under combined static and dynamic loading is formulated in a two-dimensional planar formulation based on two approaches: full-scale modeling within continuum mechanics and a simplified formulation based on the hypotheses of the theory of shells of the Timoshenko type and the Winkler foundation. Both approaches allow solving the problems of deformation and stability of non-shallow shells on the basis of Timoshenko's hypotheses, taking into account geometric nonlinearities. The statement from the perspective of continuum mechanics makes it possible to approximate the shell in thickness by a number of layers of finite elements. The constitutive relations are formulated in Lagrange variables using a fixed Cartesian coordinate system as a reference one. Kinematic relations are recorded in the metric of the current state. The elastic-plastic properties of shells are described by the theory of plastic flow with isotropic hardening. The equations of motion follow from the balance of the virtual powers of the work. In the first approach, the contact interaction of a shell and an elastic body is modeled by the conditions of nonpenetration along the normal and free slip along the tangent. The nonpenetration conditions are satisfied only in the active phase of the contact interaction; if the contact is broken, they are replaced by conditions on the free surface. In the second approach, the contact interaction of the elastic core with the shell is modeled by the Winkler foundation. Both approaches allow one to describe the nonlinear subcritical deformation of shells of revolution with an elastic core, to determine the limiting (critical) loads in a wide range of loading rates, taking into account the geometric imperfections of the shape. Using both approaches, a numerical simulation of epy contact interaction problem of an elastoplastic cylindrical shell with a thick-walled elastic core at a quasi-static uniform external pressure is carried out. The study of the influence of the thickness and initial deflection of the shell, as well as the stiffness and thickness of the core, on the value of the critical pressure and the form of buckling has been carried out. Based on these calculations, a conclusion was made about a wide range of applicability of the Winkler foundation model.

Experimental path-following of equilibria using Newton’s method. Part II: Applications and outlook

In Part I of this paper, a quasi-static experimental path-following method was developed that uses tangent quantities in a feedback controller, based on Newton’s method. The ability to compute an experimental tangent stiffness opens the door to more advanced path-following techniques. Here, we extend the experimental path-following method to: (i) pinpointing of critical points (limit and branching points); (ii) branch switching to alternate equilibrium paths; and (iii) tracing of critical points with respect to a secondary parameter. We initially explore these more advanced concepts via the virtual testing environment introduced and validated in Part I. Ultimately, the objective is to demonstrate novel testing procedures and protocols made possible by these advanced experimental path-following methods. In particular, three pertinent examples are discussed: (i) design sensitivity plots for shape-adaptive morphing structures; (ii) validation of nonlinear FE benchmark models; and (iii) non-destructive testing of subcritical (unstable) buckling of thin-walled shells.

Dent Imperfections in Shell Buckling: The Role of Geometry, Residual Stress, and Plasticity

Abstract Departures of the geometry of the middle surface of a thin shell from the perfect shape have long been regarded as the most deleterious imperfections responsible for reducing a shell’s buckling capacity. Here, systematic simulations are conducted for both spherical and cylindrical metal shells whereby, in the first step, dimple-shaped dents are created by indenting a perfect shell into the plastic range. Then, in the second step, buckling of the dented shell is analyzed, under external pressure for the spherical shells and in axial compression for the cylindrical shells. Three distinct buckling analyses are carried out: (1) elastic buckling accounting only for the geometry of the dent, (2) elastic buckling accounting for both dent geometry and residual stresses, and (3) a full elastic–plastic buckling analysis accounting for both the dent geometry and residual stresses. The analyses reveal the relative importance of the geometry and the residual stress associated with the dent, and they also provide a clear indicator of whether plasticity is important in establishing the buckling load of the dented shells.

Multi-objective optimization of sandwich composite pressure hull for decreasing weight and drag force and increasing buckling load capacity

In underwater applications, space vehicles, and aircrafts, the weight becomes an important factor. Additionally, the design of composites structures greatly depends on the number of layers and the fiber orientation angle. Therefore, this work presents the optimization of sandwich composite pressure hull in order to minimize (weight and drag force) and maximize the buckling load capacity using ANSYS Parametric Design Language (APDL). Tsai-Wu and maximum stress failure criteria were incorporated for predicting the first-ply failure. The major and minor radius of the pressure hull, the ring and long beams dimensions, the fiber orientation angle and layer thickness are taken as design variables. The results illustrated that, core thickness (Tcore ) has a great effect to resist the shell buckling. While, has a little effect on both Tsai-Wu and maximum stress failure index.

Nondestructive Prediction of the Buckling Load of Imperfect Shells

From soda cans to space rockets, thin-walled cylindrical shells are abundant, offering exceptional load carrying capacity at relatively low weight. However, the actual load at which any shell buckles and collapses is very sensitive to imperceptible defects and cannot be predicted, which challenges the of such structures. Consequently, probabilistic descriptions in terms of empirical design rules are used and designing reliable structures requires the use of conservative strength estimates. We introduce a nonlinear description where finite-amplitude perturbations trigger buckling. Drawing from the analogy between imperfect shells which buckle and imperfect pipe flow which becomes turbulent, we experimentally show that lateral probing of cylindrical shells reveals their strength nondestructively. A new ridge-tracking method is applied to commercial cylinders with a hole showing that when the location where buckling nucleates is known we can accurately predict the buckling load of each individual shell, within ±5%. Our study provides a new promising framework to understand shell buckling, and more generally, imperfection-sensitive instabilities.

(Invited) In Situ Investigation of Interface Evolution and Chemo-Mechanics in Batteries

It is critical to understand how energy storage materials transform and degrade within devices to enable the development of next-generation batteries. In my research group, multi-scale in situ techniques are used to reveal reaction mechanisms and interfacial transformations in battery materials. In this talk, I will first present our recent work on understanding and controlling transformations at interfaces between solid-state electrolytes and lithium electrodes within solid-state batteries, where we use interlinked in situ investigations to investigate how these interfacial transformations control chemo-mechanical degradation. In situ X-ray tomography experiments of operating NASICON-based cells reveal that the growth of the interphase causes fracture of the SSE, and quantification of the crack network shows that the extent of fracture with time is directly correlated to impedance increases within the cell. In situ transmission electron microscopy (TEM) shows that the reaction of LAGP with lithium is similar to a conversion reaction, in which lithium insertion causes amorphization and volume expansion of ~130%. The use of structured alloy anodes is shown to be effective for controlling degradation and improving cycle life. Finally, I will discuss results on understanding how chemo-mechanics controls the transformation process of nanocrystal-based battery anodes. Specifically, sufficiently small antimony nanocrystals with native oxide are found to undergo spontaneous and reversible hollowing during delithiation, and this process is found to be driven by the energetic balance between void formation and oxide shell buckling. Together, these findings show the importance of controlling chemo-mechanics in next-generation battery materials, with in situ experiments being critical for understanding these processes.

Global buckling and multiscale responses of fiber-reinforced composite cylindrical shells with trapezoidal corrugated cores

In order to comprehensively understand the effect of composite materials on the structural response, a novel multiscale numerical model is presented in this work to study the global buckling and localized responses of composite cylindrical shells with trapezoidal corrugated cores. Treated as hierarchical structures, the investigation of composite shells includes carbon fiber-reinforced composites at micro-scale, fiber–matrix laminates at meso-scale, as well as corrugated shells at macro-scale. The well-established locally-exact homogenization theory is firstly introduced to generate the microstructural effective coefficients, which are then transferred to the lamination theory. Finally, a simplified structural homogeneous representative volume element (RVE) is established with equivalent properties to a part of cylindrical structures with inner and outer skins embedded with corrugated cores, significantly facilitating the finite element (FE) simulations. The simplified structure is validated against the 3D full-scale analysis with good agreement for critical loads of different buckling modes. The global buckling performance of corrugated cylinders is then demonstrated by considering the effect of constituent materials, fiber volume fraction and geometric parameters. More importantly, benefited from a top-down procedure, the localized stress distributions are recovered using the three-stage multiscale model, to predict the possible crack initiations starting within the microstructures.

Buckling of boron nanotubes under axial compression: Insights from molecular mechanics and continuum mechanics

The buckling behaviours of boron nanotubes (BNTs) are investigated in this paper by using molecular structural mechanics (MSM) simulations and elastic shell models. Two types of BNTs (α and χ-BNTs) whose structures are proven to be thermally stable in first-principles molecular dynamics simulations are considered in the study. Our MSM results show a shell-to-column buckling mode transition occurring in BNTs with increasing aspect ratio. The critical buckling force of the column buckling of long BNTs significantly decreases as the aspect ratio grows, while that of the shell buckling of short BNTs is almost independent of the aspect ratio. MSM simulations also show that α and χ-BNTs both have anisotropic elastic properties. As a result, the simple Donnell shell model previously employed in other tubular nanomaterials becomes inapplicable in describing the buckling behaviours of the present BNTs. Under this circumstance, we propose an orthotropic elastic shell model to consider the anisotropy of elastic properties in the continuum mechanical modelling of BNTs. Through the orthotropic elastic shell model, the buckling behaviours of α and χ-BNTs are found to have different relations with their chirality, which is attributed to different anisotropy features observed in the elastic properties of these two BNTs.

Energy Method on Dynamic Buckling of Cylindrical Shell with Simply-Fixed Supported subjected to Axial Impact of Rigid Mass

Based on the energy method, considering the stress wave effect, the buckling problem of cylindrical shell subjected to impact is studied. Substituting the Lagrange function and the trial function satisfied the boundary condition into the second Lagrange equation. After the second-order linear partial differential equation is obtained, the analytical expression of the critical buckling speed of a rigid mass impacting cylindrical shell is obtained by analyzing the properties of the equations. The influence of critical length, impact mass, axial mode number, circumferential mode number and diameter-thickness ratio on buckling is discussed by analyzing the examples. The results show that the stress wave effect, initial kinetic energy and diameter-thickness ratio have a significant effect on the dynamic buckling of cylindrical shells; the higher-order modes of buckling and the buckling of cylindrical shell are easily motivated by high-speed impact.

A three-dimensional solution for free vibration and buckling of annular plate, conical, cylinder and cylindrical shell of FG porous-cellular materials using IGA

This study aims to present a three-dimensional (3D) numerical solution for investigating the free vibration and buckling responses of annular plate, conical and cylindrical shell made of functionally graded (FG) porous rock materials. Isogeometric analysis (IGA) is utilized in order to develop the 3D numerical solution. Furthermore, the computational package is developed using C# Programming, which is an object-oriented programming language. The distribution of porosity along the thickness direction obeys the cosine function rule, for which the top surface is porosity-free, and the bottom surface is porosity-rich. The weak form for free vibration and buckling analyses are discretized and approximated through 3D NURBS based IGA. The accuracy of the present solutions is verified by the results obtained from published data. The convergence of 3D FGP solutions is checked, and it is found that a quartic NURBS element can yield high-accuracy results with the lowest computational cost. Afterward, the numerical results showed the impact of porosity of the FGP annular plate, FGP conical, and cylindrical shell on free vibration and buckling problems.

Scale Dependent Critical External Pressure for Buckling of Spherical Shell Based on Nonlocal Strain Gradient Theory

Instabilities in nanosized, externally pressurized spherical shells are important for their applications in nano and biotechnology. Mechanics at such length scale is described by nonlocal and Strain Gradient (SG) field theories. However, analysis of shell buckling is involved and becomes even more complicated in presence of nonlocal and SG interactions. This paper demonstrates that such analysis can be largely simplified by a shallow segment representation of the shell by assuming short wave lengths for the incipient buckling modes. The governing equations are derived and linearized equations are solved to obtain a closed form solution for the critical external pressure causing buckling for a pressurized nonlocal shell. Nonlocal interactions are shown to reduce, whereas the SG interaction increases the critical pressure. The relative reduction/increase becomes more prominent for higher modes of buckling and for increasingly thinner shell. A constricting relationship between the two set of wave numbers expressing the buckling modes is also shown to be modified by the nonlocal and SG scale parameters. Consequent wave numbers increase/decrease, accompanied by decreasing/increasing number of wavelengths, thereby further justifying the shallow segment representation employed herein.

Multi-hazard failure assessment of atmospheric storage tanks during hurricanes

Hurricane as one of the most destructive natural hazards can make a devastating impact on the industrial equipment, especially atmospheric storage tanks, leading to the release of stored chemicals and disastrous safety and environmental issues. These catastrophic consequences are caused not only by strong winds but also by the torrential rainfall and inundating floods. The objective of this study is to present a risk-based methodology for assessing and reducing the vulnerability of atmospheric storage tanks to hurricanes. Considering the shell buckling, flotation, sliding, and roof sinking as dominant failure modes of atmospheric storage tanks during hurricanes, Bayesian network (BN) has been employed to combine the failure modes while considering their conditional dependencies. The probability updating feature of the developed BN was employed to indicate that the flood is the most critical hazard during hurricanes while the impact of wind and rainfall cannot be neglected. Extending the developed BN to an influence diagram, the cost-benefit filling of storage tanks with water prior to the advent of hurricanes was shown as a viable measure for reducing the damage probability. The results show that the proposed methodology can be used as an effective decision support tool for assessing and reducing the vulnerability of atmospheric storage tanks to natural hazards.

Analytical Study on Dynamic Buckling of Cylindrical Shells with General Thickness Variations Subjected to Exponentially Increasing External Pressure

This paper presents an analytical investigation on dynamic buckling of cylindrical shells with general thickness variations under exponentially increasing external pressure over the time. Different from the previous studies in literatures, the shell thickness varies arbitrarily and is common in actual engineering, which leads to failure of the existing methods. A new analytical method is first developed to solve the fourth-order governing partial differential equations with variable coefficients for the shell subjected to varying external pressure. Then the asymptotic formulae for dynamic buckling loads considering general thickness variations are derived and expressed by geometry sizes of the shell and thickness variation functions. To validate the presented results, two specific non-axisymmetric thickness cases are discussed in detail. The critical dynamic buckling loads show a great agreement with the previous ones by other researchers for simple and axial thickness variation situation. Finally, example calculations and parametric discussion are performed, and influences of thickness variation types, speed of external pressure and the power exponent of time on the critical dynamic buckling loads are discussed.

Unequivocally Nonconservative Results From One Method of Imperfection Quantification in RCC-MR

Abstract Design against buckling of thin shells at high temperatures often follows the code RCC-MR. RCC-MR allows three methods to quantify shell imperfections for use in safe load calculations, where lower imperfection values raise the safe load estimates. In recent work, we showed that the third of these methods can sometimes yield remarkably low imperfection values, leading to potentially nonconservative designs, but nonconservatism of the method was not proved. Here, we prove nonconservatism in two designs based on the third method. Proving such nonconservatism is difficult using experiments or with large material nonlinearity in simulations. We first discuss these difficulties to motivate our approach. We then present two examples: a spherical shell and a torispherical shell, both under external pressure. The shell walls are thin enough so that plasticity is not encountered before structural collapse. For specific shape imperfections, we show with geometrically nonlinear, purely elastic, highly refined, post-buckling analysis using abaqus that the physical loads at which the imperfect shells collapse are overpredicted via RCC-MR's third method by factors of about 8/7 and 11/10, respectively. We emphasize that code-based design using nonlinear simulation prescribes a further safety factor of 2.5, which we have denied ourselves here in order to give the third method the benefit of doubt. We conclude that the third imperfection quantification method in RCC-MR should be reexamined.