Imperfect Shells Research Articles

Stochastic analysis of geometrically imperfect thin cylindrical shells using topology-aware uncertainty models

Buckling of thin-shell structures is one of the most canonical problems in mechanics. In practice, the buckling load and its deviation from theoretical prediction is often handled through the development of knock-down factors in shell structure design. Uncertainty-informed analysis based design is an alternative path that has been aggressively pursued in recent years. In this study, we use a novel representation of the stochastically imperfect shell geometry to investigate the buckling behavior of imperfect cylindrical shells. The representations rest on a non-Gaussian random field model, obtained by translating a latent Gaussian field defined as the solution to a stochastic partial differential equation and allows for the construction of topology-aware spatially correlated imperfections on nonconvex domains. We perform finite element analysis of buckling for imperfect shells and gain new insights into how the interplay between random imperfections and topological features influences buckling behavior.

Analytical formulations and comparison of collapse models for risk analysis of axisymmetrically imperfect ring-stiffened cylinders under hydrostatic pressure

Risk-based design of marine pressure hulls require computationally efficient and precise models predicting collapse pressures of ring stiffened cylindrical shells as a function of realistic geometrical imperfections. However, the empirical interframe collapse models commonly implemented in design codes do not explicitly depend on imperfections, and the existing analytical models are only valid for axisymmetrically imperfect shells. The goal is to derive an analytical model that explicitly depends on axisymmetric and asymmetric imperfections. In order to derive such a model, first the stress development is investigated using the nonlinear Finite Element Analysis (FEA) of twelve marine pressure hulls having axisymmetric imperfections only. The knowledge gained from these investigations is used to qualify three collapse models. One of them, the integral model introduced by the authors, is accurate and sufficiently precise. It uses a new definition of interframe collapse, which also allows for asymmetric imperfections.

Knockdown factors for cylindrical shells caused by torsional Mode-I type geometric imperfections under axial compression

Geometrical imperfection, which is generally a result of manufacturing process and service conditions, plays a crucial role in load-bearing capacity of shell structures. This study presents a numerical study on knockdown factors of cylindrical shells as a result of torsional Mode-I type of geometric imperfections under compressive loads. The deformation patterns obtained from liner bifurcation analysis (LBA) for torsional Mode-I shape are used as a source of geometric imperfection. Then, geometrically nonlinear buckling analysis with imperfect model (GNIA) is incorporated with LBA in ANSYS Workbench to obtain limit loads of imperfect structures. A parametric study is thus performed to investigate the influence of imperfection depth on the load-bearing capacity considering a wide range of cylindrical shell configurations. Local and global buckling characteristics of the imperfect shells are examined and knockdown factors are characterized by three distinct regions as a basis of normalized imperfection depth. For a large number of shell configurations, a scattering of knockdown factors against normalized imperfection depth is given with mathematical expressions evolving lower and upper bounds. These expressions provide the minimum and maximum values of knockdown factors for a given imperfection depth, which can be treated as a design tool to ensure safety of the shell structure.

Investigation of the buckling behaviour of ring‐stiffened cylindrical shells under axial pressure

AbstractTanks and silos are often designed as ring‐stiffened cylindrical shells. Nowadays, the load‐increasing effect of tight ring stiffeners under axial compression has received just little attention in research and practice. Considering the structural safety, the ring‐stiffened shell needs to be estimated on the conservative side in practice. This is not desirable in terms of modern lightweight steel structures. Recently research results based on the numerical studies show that the axial buckling behavior of cylindrical shells can be enhanced by using ring stiffeners. In addition, the analysis of ring‐stiffened structures with numerical calculations demonstrate that cylindrical shells with ring stiffeners under axial pressure are uneconomic according to the current design standard in practice. Therefore, the results guaranteed by experiments are required, before the extensive parametric studies are putted into practice. In this paper, small specimen series are carried out to calibrate numerical models, which are subsequently used in parameter studies, because the large‐scale experiments, in the scale range of e.g. 1: 1 are cost‐intensive and therefore difficult to carry out in the laboratory. Consequently, relative small specimens with the selected radius of 796 mm and thickness 1 mm are tested. The small cylindrical shells are manufactured by welding of two half‐cylindrical shells and the ring stiffeners are spot‐welded to the cylindrical shell. The geometrical imperfection of shells is measured by a 3D scanning technologies. Finally, the experiments by means of geometrically and materially nonlinear analysis with imperfections included are recalculated on the basis of the measured data. The results illustrate that the numerical simulation agrees well with the experimental results.

Imperfection sensitivity of unstiffened cylindrical shells under external pressure

AbstractThe design rules for cylindrical shells subjected to uniform external pressure can be found in corresponding codes and recommendations, e.g. in EN 1993‐1‐6 (2007) or European Recommendation ECCS EDR5 (2008). Although the buckling strength of thin‐walled shells is highly dependent on the imperfections caused by various fabrication or manufacturing processes, the sensitivity of the buckling resistance of an imperfect shell to the amplitude of the geometric imperfections is not taken into account in design rules of standards.This paper investigates in detail the effect of boundary conditions and cylinder length on the linear buckling behaviour of cylindrical shells subjected to uniform external pressure. Following this, a comprehensive computational study is performed using geometrically nonlinear analyses on imperfect cylindrical shells with two‐imperfection types: (i) eigenmode‐affine form and (ii) longitudinal eigenmode‐affine pattern. Based on the numerical results obtained from eigenvalue analyses, the existing formulae of EN 1993‐1‐6 for the external pressure buckling factor Cβs of short cylindrical shells are modified to provide a more accurate prediction of the elastic critical circumferential buckling stress. The effects of geometric nonlinearity and imperfection sensitivity on the buckling strength are also briefly explored.

Measuring geometric imperfections of variable–angle filament–wound cylinders with a simple digital image correlation setup

Measuring the geometric imperfections in cylindrical shells is a critical step necessary to create accurate numerical models that can capture the imperfection-sensitive behavior of these structures. Modern composite structures, such as variable–angle filament–wound (VAFW) cylinders, have a unique imperfection signature that is still unknown to the scientific community. This new class of variable–stiffness structures developed by our research group combines wide tailoring capabilities with the efficient manufacturability enabled by filament winding process. The present study proposes a novel imperfection measurement method that is simple and applicable to both small and large structures. The topographic data is measured with only a pair of cameras. Practical aspects of using digital image correlation (DIC) are described and discussed in detail, such as lighting, focus adjustment, and calibration. State–of–the–art best–fit routines, based on least–squares optimization, are used to transform raw data into a common coordinate system. Finally, the transformed data is stitched to build a full 3D imperfection pattern that can be readily used in a nonlinear finite element analysis. The developed method is used to measure the imperfections of 12 VAFW cylinders. The mass of the cylinders is used to validate the geometric imperfections and evaluate the variability of the proposed methodology.

Probing the buckling of pressurized spherical shells

The prediction of the critical buckling conditions of shell structures is plagued by imperfection sensitivity. Non-destructive testing through point-load probing has been recently proposed to map the stability landscape of cylindrical shells. However, the counterpart procedure for spherical shells is still debatable. Here, we focus on the mechanical response of pressurized spherical shells containing a single dimple–like defect to a point probe. Combining experiments, finite element modeling, and existing results from classic shell theory, we characterize the nonlinear force-indentation response of imperfect shells at different pressurization levels. From these curves, we seek to identify the critical buckling pressure of the shell. In particular, the indentation angle is varied systematically to examine its effect on the probing efficacy. We find that the critical buckling point can be inferred non-destructively by tracking the maxima of the indentation force–displacement curves, if the probe is implemented sufficiently close to the defect. When probing further away from the defect, the test fails in predicting the onset of buckling since the deformation due to indentation remains localized in the vicinity of the probe. Using FEM simulations and shallow shell theory, we quantify the characteristic length associated with this localized deformation, both in the linear and nonlinear regimes. Our results demonstrate the limiting conditions of applicability for the usage of probing as a non-destructive technique to assess the stability of spherical shells.

Bi-Shell Valve for Fast Actuation of Soft Pneumatic Actuators via Shell Snapping Interaction.

Rapid motion in soft pneumatic robots is typically achieved through actuators that either use a fast volume input generated from pressure control, employ an integrated power source, such as chemical explosions, or are designed to embed elastic instabilities in the body of the robot. This paper presents a bi‐shell valve that can fast actuate soft actuators neither relying on the fast volume input provided by pressure control strategies nor requiring modifications to the architecture of the actuator. The bi‐shell valve consists of a spherical cap and an imperfect shell with a geometrically tuned defect that enables shell snapping interaction to convert a slowly dispensed volume input into a fast volume output. This function is beyond those of current valves capable to perform fluidic flow regulation. Validated through experiments, the analysis unveils that the spherical cap sets the threshold of the snapping pressure along with the upper bounds of volume and energy output, while the imperfect shell interacts with the cap to store and deliver the desired output for rapid actuation. Geometry variations of the bi‐shell valve are provided to show that the concept is versatile. A final demonstration shows that the soft valve can quickly actuate a striker.

Flutter of geometrical imperfect functionally graded carbon nanotubes doubly curved shells

In this paper, flutter of geometrical imperfect functionally graded carbon nanotubes (FG-CNTs) doubly curved shell subjected to a supersonic flow is investigated. For this purpose, the imperfect doubly curved shell is reinforced by four different carbon nanotubes (CNTs) distributions in the thickness direction. The Hill’s elastic moduli are used to obtain the effective material properties of FG-CNTs doubly curved shell. The piston aerodynamic theory and von Karman geometrical nonlinearity terms are used to study large deflection flutter analysis of FG-CNTs imperfect shell. The effect of different parameters including large deflection, geometrical imperfection and CNTs volume fraction on critical flutter pressure of doubly curved shells are studied. According to the results, distributions with poor CNTs in the middle of thickness are more efficient in improving critical flutter pressure. Additionally, the effect of shell imperfection on flutter pressure is more considerable than CNTs volume fraction and distributions.

Experimental investigation of buckling of dented cylindrical shells under axial compression

The results of unique experimental studies of buckling of dented axially compressed unstiffened cylindrical shells are provided in the suggested paper. The dimple-like geometrical imperfections of different shape, size, deflection amplitude, and number of dents were created on cylindrical shells before axial loading. The initial deflection amplitude varied in the wide range, up to several shell thicknesses. In all cases the plateaus were discovered on the buckling load initial deflection diagrams. Corresponding axial loads were close to local lower buckling loads of perfect shells. These loads are defined as minimum values of axial force when the post-buckling equilibrium states of the perfect structure with one or several dimples exist. It was shown that the lower local buckling loads slightly depend on geometrical imperfections and therefore can be used for estimation of design buckling load of the structure. Over 50 specimens were tested. Pre-buckling and post-buckling behavior of imperfect shells was studied.

Energy barrier method for estimation of design buckling load of axially compressed elasto-plastic cylindrical shells

Energy barrier as a measure of metastability of axially compressed cylindrical shell is developed in the case of elasto-plastic material of the structure. The concept of local shell buckling is used for estimation of design buckling load. It does not require measuring initial geometrical shell imperfections which is challenging or sometimes impossible in the case of large scale real structures. Parametric analysis of the structure is performed and corresponding formula is derived for design buckling load which splits up the zone of high sensitivity of the shell to imperfections. The obtained formula is compared with published experimental results and Eurocode EN 1993-1-6 (2007) recommendations.

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.

Static and dynamic postbuckling analysis of imperfect SSFG cylindrical shells surrounded by nonlinear elastic foundation subjected to an axial compression

In this article, the semianalytical method based on the Galerkin technique and fourth-order Runge-Kutta method is utilized to investigate the static and dynamic postbuckling analysis of internal/external spiral stiffened functionally graded (ISSFG/ESSFG) cylindrical shells, respectively. The simply supported SSFG shell subjected to an axial compression is resting on a nonlinear elastic foundation (EF) which is including the Pasternak and Winkler foundation parameters augmented by a softening/hardening cubic nonlinearity. The material constitutes of both the stiffeners and the shells that are continuously changed along the thickness. The discretized motion equation is extracted utilizing the Galerkin method regarding to the Donnel’s shells theory and nonlinear strain-displacement von Kármán. To obtain the responses of dynamic postbuckling (DPB) analysis, the fourth-order Runge-Kutta method is utilized. To validate the results, comparisons are made with the available solutions for both static postbuckling (SPB) and DPB analysis of stiffened homogeneous and functionally graded cylindrical shells. The influences of various material and geometrical parameters with the inclusion of the supporting nonlinear EF on the SPB and DPB behavior of imperfect ISSFG/ESSFG cylindrical shells are presented. As one of the most interesting results, for the ISSFG cylindrical shells with the stiffeners angle and ESSFG cylindrical shells with the stiffeners angle the buckling load-bearing capacity is the most.

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.

Influential geometric imperfections in buckling of axially compressed cylindrical shells – A novel approach

Selection of influential geometric imperfections may be crucial for efficient calculations of lower buckling strengths of thin cylindrical shells subject to axial compression by the geometrically/and materially nonlinear FEM analysis with imperfections (GNIA/GNMIA). The imperfections in the shapes of eigenmodes provided by the preliminary linear buckling analysis (LBA) of perfect shells are brought into focus. For categorizing a large set of potentially influential eigenmodes customarily normalized to unit amplitude the strain energy of the shell deformed in the shape of each eigenmode is determined. By the square root of the strain energy an unconventional energy measure of the shell geometric imperfections is defined. Considering the eigenmode imperfections and localized imperfections normalized by the amplitude and by the energy measure the buckling strengths of an imperfect shell are compared. A preference is given to the normalization of the geometric imperfections by the energy measure. Nevertheless, for the currently revised Structural Eurocodes, particularly for the standardization of GNIA/GNMIA FEM design of shells based on amplitudes of geometric imperfections an amendment is suggested.

Fast buckling load numerical prediction method for imperfect shells under axial compression based on POD and vibration correlation technique

Vibration correlation technique (VCT) is an effective non-destructive buckling experimental technique for shell structures. In this study, VCT is studied from the point-of-view of being a buckling load numerical prediction method by numerically simulating the experimental procedure of VCT. Firstly, the formulas of VCT are introduced for axially loaded cylindrical shells and conical shells under the clamped–clamped boundary condition. According to the VCT formulas, the numerical procedure of VCT is provided. In order to accelerate the repeated eigenvalue analysis of VCT, the proper orthogonal decomposition (POD) method is integrated into VCT, and the POD-VCT is developed. Extensive examples are presented to verify the effectiveness of the proposed method, including unstiffened cylindrical shell with real measured imperfection, unstiffened conical shell with single perturbation load imperfection, composite cylindrical shell with eigenmode imperfection, and hierarchical stiffened cylindrical shell with combined imperfection. In comparison to buckling test results, high-fidelity explicit dynamic method and VCT method, the high prediction accuracy and efficiency of the proposed POD-VCT are fully demonstrated. Additionally, example results indicate the strong applicability of the proposed POD-VCT for various types of structural configurations, materials and imperfections. Above all, the POD-VCT is verified to be a fast buckling load numerical prediction method for imperfect shells.

Effect of geometric imperfections on anchor loss and characterisation of a gyroscope resonator with high quality factor

A critical functional part of a hemispherical resonator gyroscope (HRG) is the mechanical resonator, and a few million quality factor (Q-factor) is needed for the lowest resolution. This paper focuses on anchor loss of a HRG of a few millimeters in size. A detailed parametric study of dimensions and shell imperfections due to fabrication is carried out. A sensitivity study of the effect of shell mean radius, shell thickness, stem radius, stem height on the Qanchor is carried out. The effect of geometric imperfections such as shell offset, shell tilt, shell thickness variation, and unbalance is studied in detail. From the study, it is inferred that the anchor loss becomes very significant and approaches other loss mechanisms even with minor geometric imperfections in the hardware realisation. Based on the sensitivity study, the dimensional and geometric tolerances are arrived for precision fabrication. Precision resonator is fabricated as per the requirement of minimum anchor loss. The significance of other damping mechanisms such as air damping, excitation-induced damping, thermoelastic dynamic damping and surface dissipation is also discussed. Surface characterisation before and after surface treatment has been carried out using nanoindentation technique with regard to surface loss. Functional parameters of operating frequency and Q-factor are evaluated using laser Doppler vibrometry (LDV).

Buckling of cylindrical shells subjected to a finite number of lateral loads: application to single-walled carbon nanotubes

The continuous loading assumption is adopted in most present studies on the buckling of thin shells subjected to lateral loads, while the relationship between the finite number of loads and the lateral buckling remains unclear. In this work, we derive an analytic formula for the dependence of the critical buckling stress on the number of loads, which shows that the critical stress increases significantly with the increase of the load number and reaches a saturation value in the limit of large load number. Furthermore, the analytic formula reveals the dependence of the critical stress on the deviation of the radius in an imperfect shell. To verify the validity of the analytic formula, we perform molecular dynamics simulations to investigate the buckling of both perfect and imperfect single-walled carbon nanotubes under a finite number of lateral loads, where the analytic formula agrees with the simulation results. These results shall be valuable for understanding mechanical stability of elastic thin shells or nanoscale tubal structures subjected to discrete lateral loads.

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.

Porosity effects on post-buckling behavior of geometrically imperfect metal foam doubly-curved shells with stiffeners

This papers studies nonlinear stability and post-buckling behaviors of geometrically imperfect metal foam doubly-curved shells with eccentrically stiffeners resting on elastic foundation. Metal foam is considered as porous material with uniform and non-uniform models. The doubly-curved porous shell is subjected to in-plane compressive loads as well as a transverse pressure leading to post-critical stability in nonlinear regime. The nonlinear governing equations are analytically solved with the help of Airy stress function to obtain the post-buckling load-deflection curves of the geometrically imperfect metal foam doubly-curved shell. Obtained results indicate the significance of porosity distribution, geometrical imperfection, foundation factors, stiffeners and geometrical parameters on post-buckling characteristics of porous doubly-curved shells.