Buckling Of Shells Research Articles

Sensitivity and uncertainty propagation in buckling of spherical shells under external pressure

The simple analytical model of spherical shell buckling suggested in the previous publication in the TWS journal is applied to studying the response of the structure to different perturbations given as random variables. Among them probing radial force, energy barrier and geometrical imperfections are considered as factors decreasing uniform buckling pressure. The model is expanded to the case of imperfect shells. The propagation of uncertainty of perturbations is studied by estimation the standard deviation of the buckling load. Monte Carlo method and the propagation of error formula based on Tailor expansion are used. The application of the methods to design buckling pressure estimation is discussed.

Elastic Buckling of Oblate Hemi-Ellipsoidal Shells Subjected to Hydrostatic Pressure

This study investigates the elastic buckling behavior of oblate hemi-ellipsoidal shells (OHES) subjected to non-uniform external hydrostatic pressure. The virtual work-energy of OHES consists of virtual strain energy due to membrane, bending, in-plane stress resultants, and virtual work of hydrostatic pressure. For buckling analysis, the geometric stiffness matrix is obtained from the strain energy due to in-plane stress resultants. A finite element procedure via a C1 continuity axisymmetric element is applied to solve the critical hydrostatic pressure from the eigenvalue buckling problem. The buckling pressure of the hemi-ellipsoidal dome subjected to uniform external pressure is verified with the previous research and FEM commercial software. Present results also indicate that the maximum in-plane stress of the hemi-ellipsoidal shell shapes are near the apex point as the axisymmetric buckling shape. In the case of hydrostatic pressure, the critical hydrostatic pressure of OHES is determined and are in good agreement when compared with the experimental results in published research. Furthermore, the shape ratio influences the difference in critical load results between uniform pressure and hydrostatic pressure, especially when the shape ratio is higher than 0.5 and the [Formula: see text] ratio is less than or equal to 100. Seawater depth limitations in subsea engineering are also presented and found that the shell thickness and shape ratio are the major factors affecting critical seawater depth.

Dynamic stability of a hemispherical shell with shape imperfections

The nonlinear dynamic analysis of imperfect hemispherical shell under pressure was executed. The finite element model of hemisphere in the software NASTRAN was built. The shell wall in the form of the three-cornered and four-cornered finite element net was presented. Shape imperfection as a lower buckling form (Buckling) of perfect hemispherical shell under action of static pressure was modelled. Value of imperfection amplitude was set proportionally to a shell wall thickness. Two boundary conditions in the form rigid and hinged supports on the nodes of lower edge of the shell were considered. Excitation as external pressure, which linearly depended on time and uniform distributed on elements of hemispherical shell was presented. The modal analysis of perfect hemisphere with different wall thicknesses and shell with modelled shape imperfections by the Lanczos method using computational procedure of task on natural vibrations (Normal Modes) was executed. The nonlinear dynamic analysis (Nonlinear Direct Transient) of imperfect hemispherical shell under pressure by N’yumark method was executed. Influence of modelled shape imperfections amplitude on the critical values of dynamic loading and appropriate deformation forms of shell with different boundary conditions were investigated.
 It was discovered that a modelled imperfection in the form of a lower buckling form of perfect shell under static pressure in the modal and the dynamic analysis of hemisphere under the same type of the loading was effective. A significant influence of the amplitude of the shape imperfection on the critical values of the dynamic load (55 %) was observed. Also, in our opinion, the presented model of shape imperfection can be used to assess design reliability of the hemisphere under the action of dynamic loads using Bolotin probabilistic approach.

Defect-Defect Interactions in the Buckling of Imperfect Spherical Shells

Abstract We perform finite element simulations to study the impact of defect-defect interactions on the pressure-induced buckling of thin, elastic, spherical shells containing two dimpled imperfections. Throughout, we quantify the critical buckling pressure of these shells using their knockdown factor. We examine cases featuring either identical or different geometric defects and systematically explore the parameter space, including the angular separation between the defects, their widths and amplitudes, and the radius-to-thickness ratio of the shell. As the angular separation between the defects is increased, the buckling strength initially decreases, then increases before reaching a plateau. Our primary finding is that the onset of defect-defect interactions, as quantified by a characteristic length scale associated with the onset of the plateau, is set by the critical buckling wavelength reported in the classic shell-buckling literature. Beyond this threshold, within the plateau regime, the buckling behavior of the shell is dictated by the largest defect.

A modified analytical model for predicting seismic behaviors of unanchored liquid storage tanks

Large-scale liquid storage tanks, which are generally constructed unanchored, would suffer from severe damages subjected to intensive earthquakes, e.g., uplift, shell buckling and piping connection fracture, or even failure. This paper aims to develop a modified analytical model for performance assessment of large-scale unanchored storage tanks subjected to earthquakes, with a special focus on the effect of the uplift behavior. The stiffness of the rotational spring at the base of the analytical model was derived from an artificial neural network (ANN) on the spring resistance function. For this purpose, finite element (FE) models of unanchored storage tanks for static analysis were first verified by using experimental and numerical results from literature. Subsequently, the static analysis was performed on a large-scale unanchored storage tank to obtain the spring resistance function, i.e., the overturning moment – uplift height relationship, and a parametric analysis was further developed. Accordingly, a database consisted of 324 FE models was generated to establish the ANN and the spring resistance function was consequently derived. Responses of the large-scale storage tank under different earthquakes obtained from the modified model were compared with the numerical simulation result and reasonable accuracy with deviations of −24% and − 14% on the peak uplift height and peak base shear force was witnessed, respectively.

Predicting buckling of carbon fiber composite cylindrical shells based on backpropagation neural network improved by sparrow search algorithm

Predicting buckling of carbon fiber composite cylindrical shells based on backpropagation neural network improved by sparrow search algorithm

Effect of Localized Defect Positioning on Buckling of Axisymmetric Cylindrical Shells under Axial Compression

Effect of Localized Defect Positioning on Buckling of Axisymmetric Cylindrical Shells under Axial Compression

Nonlinear ultrasound in liquid containing multiple coated microbubbles: effect of buckling and rupture of viscoelastic shell on ultrasound propagation

With promising applications in medical diagnosis and therapy, the behavior of shell-encapsula-ted ultrasound contrast agents (UCAs) has attracted considerable attention. Currently, second-generation contrast agents stabilized by a phospholipid membrane are widely used and studies have focused on the dynamics of single phospholipid shell-encapsulated microbubbles. To improve the safety and the efficiency of the methods using the propagation or targeted ultrasound, a better understanding of the propagation of ultrasound in liquids containing multiple encapsulated microbubbles is required. By incorporating the Marmottant–Gompertz model into the multiple scale analysis of two-phase model, this study derived a Korteweg–de Vries–Burgers equation as a weakly nonlinear wave equation for one-dimensional ultrasound in bubbly liquids. It was found that the wave propagation characteristics changed with the initial surface tension, highlighting two notable features of the phospholipid shell: buckling and rupture. These results may provide insights into the suitable state of microbubbles, and better control of ultrasound for medical applications, particularly those that require high precision.

Analytical investigation on the buckling of cylindrical shells under combined non-uniform axial compression and external pressure

This paper focuses on the buckling problem of cylindrical shells under combined non-uniform axial compression and external pressure. To obtain buckling loads, a completely new method called two-parameter perturbation method is established. Using this method, governing differential equations are solved and buckling loads expressed by load functions, load parameters, and shell dimensions are derived for the first time. Comparative studies show that buckling loads by the analytical formulas coincide with those for combined uniform loads or single wind load in literatures. Buckling of a cylinder subjected to combined bending and wind loads is analyzed, and confirmed by the Galerkin method for the case of small load variations. Therefore, the presented formulas are adequately validated. The numerical calculations are carried out to analyze the effects of non-uniform load parameters on buckling load coefficient. The obtained results in this paper can be applied to evaluate buckling loads for cylinders subjected to combined non-uniform loads in actual engineering.

Effect of potting support design on compression buckling of composite cylindrical shells

The design of thin-walled cylindrical shells under compression loading is mostly driven by buckling considerations. However, accurate experimental evaluation of the buckling load is challenging due to small variations in boundary conditions due to manufacturing tolerances in support conditions. To test a cylindrical shell under compression loading, potting support is usually created around its bottom circumference to avoid edge crippling that could otherwise drastically reduce its critical buckling load. Therefore, investigating the effects of the potting support on the buckling response of cylindrical shell structures is important to mimic boundary conditions as close as possible to real structural constraints of the boundary. The main objective of this work is to investigate the effects of single and double-sided potting supports on the critical linear buckling load of composite cylindrical shells under compression loading. Then, this paper provides insights into understanding underlying reasons for the deviations of theoretical buckling loads from their corresponding experimental values due to boundary conditions, which can occur independently from, or in combination with, geometric imperfection sensitivity. Finally, robust linear buckling expressions that can help designers estimate the reduction due to support conditions are presented. These expressions can be used for evaluating initial safety margins for the potting support design used for testing cylindrical shells.

Buckling analysis of thick cylindrical shells using micropolar theory

Modeling complex materials with internal structure such as porous solids is challenging as in some cases the classical theory cannot provide precise responses. By considering the scale effects through additional kinematic descriptors and six constants for isotropic materials, the micropolar theory can accurately model complex materials like bone. This paper studies the buckling of a thick cylindrical shell using classical and non-classical theories. The cylinder’s material is considered bone and isotropic, and the critical load value for different geometries and boundary conditions has been obtained. Finally, the size-effect and importance of micropolar theory in micro dimensions are investigated. The micropolar equations are solved by a numerical solution using the 3D finite element method. The results show that decreasing the macroscopic length increases the stiffness of the cylinder more than that predicted by classical theory; In addition, by increasing the thickness of the cylinder and the importance of shear stresses, the micropolar theory predicts a more critical load than the classical one, and the result differences become more significant between micropolar and classical theories. Also, the characteristic length of the micropolar is investigated. The results show that the change of the critical load increased by moving toward the micro dimensions.

Properties of local buckling of spherical shell under external pressure

An important property of localization in buckling of spherical shells under external pressure is discussed. It is shown that the localization is possible for structures with nonlinear softening. An analytical model of local buckling of spherical shell is developed. Rayleigh–Ritz method is used at small and moderate deflections. At large deflections corresponding to relatively small pressure (less than 20% of classical bucking load) it is based on asymptotic method. The asymptotic model is then expanded to the practically important range of the load. The response of the structure to local perturbations of different types (including radial probing force, prescribed deflection at the shell pole, and energy barrier) is studied. Special attention is paid to the energy barrier which is required for structure transition from initial equilibrium state to the post-buckling dimple-like state. Energy barrier criterion is used as a measure of metastability of the structure and applied for estimation of load level separating high and low sensitivity of the shell to local perturbations. Based on this pressure value, formulae for design buckling load are proposed and deliberated. Similarities and differences of local buckling of spherical shells under external pressure and axially compressed cylindrical shells are discussed.

Foreword to the special issue on new developments in structural stability.

Foreword to the special issue on new developments in structural stability.

Probabilistic buckling of imperfect hemispherical shells containing a distribution of defects.

The buckling of spherical shells is plagued by a strong sensitivity to imperfections. Traditionally, imperfect shells tend to be characterized empirically by the knockdown factor, the ratio between the measured buckling strength and the corresponding classic prediction for a perfect shell. Recently, it has been demonstrated that the knockdown factor of a shell containing a single imperfection can be predicted when there is detailed a priori knowledge of the defect geometry. Still, addressing the analogous problem for a shell containing many defects remains an open question. Here, we use finite element simulations, which we validate against precision experiments, to investigate hemispherical shells containing a well-defined distribution of imperfections. Our goal is to characterize the resulting knockdown factor statistics. First, we study the buckling of shells containing only two defects, uncovering non-trivial regimes of interactions that echo existing findings for cylindrical shells. Then, we construct statistical ensembles of imperfect shells, whose defect amplitudes are sampled from a lognormal distribution. We find that a 3-parameter Weibull distribution is an excellent description for the measured statistics of knockdown factors, suggesting that shell buckling can be regarded as an extreme-value statistics phenomenon. This article is part of the theme issue 'Probing and dynamics of shock sensitive shells'.

Pseudo-bistability of viscoelastic shells.

Viscoelastic shells subjected to a pressure loading exhibit rich and complex time-dependent responses. Here we focus on the phenomenon of pseudo-bistability, i.e. a viscoelastic shell can stay inverted when pressure is removed, and snap to its natural shape after a delay time. We model and explain the mechanism of pseudo-bistability with a viscoelastic shell model. It combines the small strain, moderate rotation shell theory with the standard linear solid as the viscoelastic constitutive law, and is applicable to shells with arbitrary axisymmetric shapes. As a case study, we investigate the pseudo-bistable behaviour of viscoelastic ellipsoidal shells. Using the proposed model, we successfully predict buckling of a viscoelastic ellipsoidal shell into its inverted configuration when subjected to an instantaneous pressure, creeping when the pressure is held, staying inverted after the pressure is removed, and eventually snapping back after a delay time. The stability transition of the shell from a monostable, temporarily bistable and eventually back to the monostable state is captured by examining the evolution of the instantaneous pressure-volume change relation at different time of the holding and releasing process. A systematic parametric study is conducted to investigate the effect of geometry, viscoelastic properties and loading history on the pseudo-bistable behaviour. This article is part of the theme issue 'Probing and dynamics of shock sensitive shells'.

Buckling of lipidic ultrasound contrast agents under quasi-static load.

Collapse of lipidic ultrasound contrast agents under high-frequency compressive load has been historically interpreted by the vanishing of surface tension. By contrast, buckling of elastic shells is known to occur when costly compressible stress is released through bending. Through quasi-static compression experiments on lipidic shells, we analyse the buckling events in the framework of classical elastic buckling theory and deduce the mechanical characteristics of these shells. They are then compared with that obtained through acoustic characterization. This article is part of the theme issue 'Probing and dynamics of shock sensitive shells'.

Hitting the mark: probing at the initiation site allows for accurate prediction of a thin shell's buckling load.

Geometric imperfections are understood to play an essential part in the buckling of a thin shell, but how multiple defects interact to control the onset of failure remains unclear. Here, we examine the failure of real cylindrical shells by experimentally poking soda cans with a large imparted dimple. By high-speed imaging of the can's surface, the initiation of buckling from axial loading is directly observed, revealing that larger dimples tend to set the initial buckling location. However, the influence of the shell's background geometric imperfections can still occasionally dominate, causing initiation to occur far from the dimple. In this situation, probing at the dimple leads to an over-prediction of the axial capacity. Using finite-element simulations, we understand our experimental results as a competition between the large dimple and the shell's inherent defect structure. In our simulations, we empirically observe a deformation-based criterion that connects the ideal poking location to the initiation site. This article is part of the theme issue 'Probing and dynamics of shock sensitive shells'.

The Effects of Favorable and Unfavorable Geometric Perturbations on the Response of Pressure-Loaded Hemispherical Shells

Abstract Hemispherical shells subjected to external pressure loading are known to be sensitive to geometric imperfections. Dimple imperfections and their effects on the buckling and postbuckling response of spherical and hemispherical shells under externally applied pressure loads have been widely studied. The studies have shown that dimple imperfections are unfavorable, and their presence leads to a drastic lowering of buckling pressure, the severity being dependent on the radius-to-wall thickness ratio of the shell. Motivated by a plenary lecture presented by Hutchinson, we have conducted equilibrium analysis through finite element computations of externally pressurized hemispherical shells to understand if we can intentionally design shells with initial geometric perturbations that are favorable to resisting external pressure. We have studied dimple imperfections that can either increase or decrease the local curvature. We show that outward pointing dimples outperform inward-pointing dimples in such a structure and hence can be viewed as being favorable with regards to shell buckling under external pressure. With advances in precision manufacturing, the results presented here serve as a guide to designing shells with intentional perturbations to the initial shell geometry that can lead to favorable outcomes.

Nonlinear Analytical Investigation of Dynamic Buckling of a Spherical Shell Trapped under a Periodic Load

Nonlinear Analytical Investigation of Dynamic Buckling of a Spherical Shell Trapped under a Periodic Load

Reliability analysis and design of randomly imperfect thin cylindrical shells against post-critical drops

Thin-walled cylindrical shells are employed for high specific bending stiffness and light design. Elastic buckling of shell is a critical design consideration. The Knock Down Factors (KDFs) are stipulated for accounting the markedly reduced experimental critical load by multiplying the linearized critical load by the KDF. Key to such reductions is the nonlinearity induced modal interactions among the closely spaced buckling modes, over which, the geometric imperfections act in synergy. Being the lower bound of an extensive experimental dataset; the stipulated design KDFs are largely over-conservative; which can be relaxed by accounting for imperfections. With this as the eventual goal, this study first demonstrate the feasibility of modeling the KDFs as uniformly distributed Uncertain-but-Bounded (UBB) type variable. The UBB description is employed to obtain the reliability bounds and the design variables (radius-to-thickness R/t and L/R ratio) for achieving a target reliability index through Reliability Based Design (RBD). Repeated computations of KDFs for reliability assessment are bypassed by fitting the KDFs vs. R/t relations for given L/R. The approach is demonstrated on a thin cylindrical shell with a zero-mean, stationary, gaussian, stochastic imperfection fields. The Riks method is employed for the nonlinear finite element analysis of the shell using the code ABAQUS. The KDFs and reliability bounds are provided for varying imperfections and are compared with an existing study on refining the KDFs. Furthermore, the RBD presents the desired R/t for a given L/R for achieving a target reliability index and the respective KDF for a given imperfection level. The L/R dependency is shown to reduce the over-conservatism in the design stipulation.