Dimple Imperfection Research Articles

Simulating the effects of geometric imperfections on the buckling of axially compressed cylindrical shells through reducing localized stiffness

Thin-walled cylindrical shells are prone to buckling under axial compression. The load-carrying capacity of the shells can thus be significantly reduced. It's well known that the buckling loads of axially compressed cylindrical shells are quite sensitive to the initial geometric imperfections. Even though the magnitudes of these imperfections are relatively small, the corresponding buckling loads can be significantly reduced compared to the perfect shells. In this paper, the effects of geometric imperfections on the buckling loads are equivalently simulated by reducing localized stiffness of the cylindrical shells. Firstly, the method for simulating the effects of geometric imperfections on the buckling loads of cylindrical shells is introduced in detail. Then, the proposed method is used to simulate the effects of single dimple imperfection and multiple dimple imperfections on the buckling loads of cylindrical shells. Results show that the buckling loads and load-displacement curves of the shells with single and multiple dimple imperfections can both be well simulated by the localized stiffness reduction method. Furthermore, the measured geometric imperfections from manufactured cylindrical shells are investigated. The numerical results simulated by the proposed method are also verified by the shell models with real measured geometric imperfections and the experimental results from the literature. The applicability and reliability of the proposed equivalent method in dealing with complex imperfection configurations can thus be validated. Results show that the proposed equivalent method is suitable for simulating various geometric imperfections of the cylindrical shells. It is promising to be applied as an efficient tool to quantify the imperfection sensitivity of buckling loads and provide a new solution for buckling analysis of cylindrical shells.

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

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

Experimentally probing the stability of thin-shell structures under pure bending.

This paper studies the stability of space structures consisting of longitudinal, open-section thin-shells transversely connected by thin rods subjected to a pure bending moment. Localization of deformation, which plays a paramount role in the nonlinear post-buckling regime of these structures and is extremely sensitive to imperfections, is investigated through probing experiments. As the structures are bent, a probe locally displaces the edge of the thin shells, creating local dimple imperfections. The range of moments for which the early buckling of the structures can be triggered by this perturbation is determined, as well as the energy barrier separating the pre-buckling and post-buckling states. The stability of the local buckling mode is then illustrated by a stability landscape, and probing is extended to the entire structure to reveal alternate buckling modes disconnected from the structure's fundamental path. These results can be used to formulate efficient buckling criteria and pave the way to operating these structures close to their buckling limits, and even in their post-buckling regime, therefore significantly reducing their mass. 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.

Buckling design of stiffened cylindrical shells under axial compression based on energy barrier approach

Buckling design of axially compressed cylindrical shells is still a challenging subject due to the well-known imperfection sensitivity characteristic. Compared to the unstiffened ones, design of buckling loads for the stiffened cylindrical shells composed of skins and stiffeners is more difficult considering about the complicated effects of structural parameters on the imperfection sensitivity. In this paper, the promising buckling design method based on energy barrier approach is applied to the axially compressed stiffened cylindrical shells. Three manufactured stiffened cylindrical shells from references are selected to verify the reliability of the method by comparing with the existing experimental data. In addition, various stiffened cylindrical shell models with dimple imperfections are designed and numerically researched in this paper to further prove the reliable design buckling loads provided by this method. The buckling loads designed by other similar methods are also taken for comparison. Results have verified the reliability and advantage of the method in this paper for clarifying imperfection sensitivity and designing buckling loads of stiffened cylindrical shells. Then, using this method, the complicated effects of different structural parameters on the imperfection sensitivity and design buckling loads of the stiffened cylindrical shells are systematically investigated. Based on that, the stiffened cylindrical shells with both high design buckling loads and quite low imperfection sensitivity which are always favorable in lightweight design have been successfully obtained. It also indicates a potential improvement for the load-carrying efficiency of current stiffened cylindrical shells using the buckling design method based on energy barrier approach.

On the imperfection sensitivity and design of tori-spherical shells under external pressure

Tori-spherical shells are often used as enclosures of pressure vessels in ocean and civil engineering. When subjected to external pressure, these thin-walled shells are prone to buckling. The corresponding critical buckling pressure heavily depends on deviations from the ideal shell shape, but also the yield strength as well as the knuckle radius.This article summarizes and analyzes all known experimental results for tori-spherical shells under external pressure. A detailed numerical elastic-plastic buckling analysis of a tori-spherical bulkhead is presented including details regarding non-linear material model, finite-element modeling and solver settings.In addition, a wide variety of empirical design and numerical geometric imperfection approaches for tori-spheres are presented, applied and validated. Among the geometric imperfection approaches are realistic measured geometric imperfections, dimple imperfections, flat patch imperfections and reduced stiffness methods. Large scale parametric imperfection amplitude analyzes are performed in order to determine which imperfection concept delivers in general the best fit to experimental results.New design factors for different tori-spherical shell geometry configurations are consequently developed and validated with experimental results. The new design factors lead to significantly improved critical load estimations in comparison to lower-bounds obtained empirically.

Study on the Axial Compression Postbuckling Similitude Model of the Stiffened Cylindrical Shell with Dimple Imperfections

The body of the new‐type dry gas holder is a large stiffened cylindrical shell. Limited by the test site and economic conditions, the buckling characteristics of such holders are generally studied through scale model experiments. Taking the longitudinal‐ring rectangular stiffened cylindrical shell as the research object, the generalized similitude condition and scaling principle formula of the structure are derived innovatively based on Donnell’s assumption and the energy method. By means of displacement loading and node coordinates updating, dimple imperfections are introduced into the ideal structure of the stiffened cylindrical shell, and then, the complete similitude and partial similitude analysis of axial compression nonlinear buckling for imperfect structures are carried out. The analysis results show that the complete similitude analysis of stiffened cylindrical shell axial‐compression nonlinear buckling can be realized accurately; the partial similitude model for stiffened cylindrical shell axial‐compression nonlinear buckling can better predict the buckling characteristics of its prototype structures, and the closer the Poisson’s ratio between the model and the prototype materials is, the more accurate the prediction results are. Meanwhile, the generalized similitude condition and scaling principle formula derived based on the energy method can provide useful reference for the model design and experimental verification of the axial compression buckling of the stiffened cylindrical shell with local geometric imperfections.

Collapse of Conical Shells Having Single Dimple Imperfection Under Axial Compression

Abstract This paper aims to provide experimental results into the buckling behavior of cones having single dimple imperfection subjected to axial compression. Results of eight laboratory scaled conical test models and their accompanying numerical data are presented. Cones were manufactured in pairs with single dimple imperfection amplitude, A, of 0.0, 0.56, 1.12, and 1.68. Experimental results reveal good repeatability of collapse load. The errors between each pair were found to be 3%, 7%, 11%, and 1%. Furthermore, the comparison between test data and the numerically predicted collapse load was seen to be good. The ratio of collapse test load to finite element predicted values are [(0.96, 0.99), (1.04, 1.10), (1.06, 0.94), (1.0, 1.01)]. The buckling of axially compressed conical shells using the single dimple imperfection type appears to be strongly influenced by: (i) the dimple/imperfection amplitude, (ii) the cone geometric parameter, i.e., radius-to-thickness ratio and cone angle, and (iii) the location of the dimple.

Lower-bound axial buckling load prediction for isotropic cylindrical shells using probabilistic random perturbation load approach

Due to high sensitivity to various imperfections, buckling loads of thin-walled cylindrical shells subjected to axial load vary dramatically. In order to predict lower-bound buckling loads for axially loaded cylindrical shells rationally, a probabilistic analysis approach named Probabilistic Random Perturbation Load Approach (PRPLA) is developed in this study. Firstly, a Back-Propagation Neural Network (BPNN) based method is established to describe measured imperfection patterns. Next, Random Single Perturbation Load Approach (RSPLA) is loaded upon BPNN-based depicted traditional imperfection patterns to construct a stochastic dimple imperfection. The aforementioned scattering traditional imperfections, as well as a variety of scattering non-traditional imperfections, are then sampled using Monte-Carlo simulation to generate cylindrical shell models differentiating from a nominal one. The probabilistic distribution of lower-bound buckling loads is obtained by finite element analysis. A nominal shell's realistic lower-bound buckling load is determined by choosing a specified reliability level lastly. The results show that describing measured imperfection patterns via BPNN is very close to real ones, and PRPLA presented is an improved method to find lower-bound buckling loads efficiently compared with NASA SP-8007 and many commonly used numerical approaches.

Experimental and numerical collapse properties of externally pressurized egg-shaped shells under local geometrical imperfections

The present investigation is devoted to the nonlinear elastic collapse properties of externally pressurized egg-shaped shells under local geometrical imperfections. The shells have nominal major axis, nominal minor axis, and nominal wall thickness of 256 mm, 180 mm, and 2 mm, respectively. Nine resin egg-shaped shells are manufactured, measured accurately, tested hydrostatically, and analysed numerically. Of these, three each have nominally identical perfect geometries, nominally identical geometries with local flat imperfections, and local dimple imperfections. Furthermore, the effect of local and eigenmode imperfection depths on the collapse pressures of egg-shaped shells are studied numerically using nonlinear elastic buckling analysis. Both experimental and numerical data are presented and compared through graphs and tables.

Buckling of imperfect spherical caps with fixed boundary under uniform external pressure

Astract This study examined the buckling of spherical caps with four different geometric imperfections: local inward dimple, increased-radius, force-induced dimple, and linear buckling mode. The influence of imperfection amplitude, meridional position, and meridional extent on cap buckling was numerically explored and partially validated using experiments. The numerical and experimental data were well consistent. Results indicated that, in the case of small-sized imperfections, the linear buckling mode-shaped imperfection presented a relatively conservative cap buckling prediction compared with those of the other three imperfections. This finding contradicts previous findings that the force-induced dimple imperfection is the most unstable imperfection.

On Establishing Buckling Knockdowns for Imperfection-Sensitive Shell Structures

This paper investigates issues that have arisen in recent efforts to revise long-standing knockdown factors for elastic shell buckling, which are widely regarded as being overly conservative for well-constructed shells. In particular, this paper focuses on cylindrical shells under axial compression with emphasis on the role of local geometric dimple imperfections and the use of lateral force probes as surrogate imperfections. Local and global buckling loads are identified and related for the two kinds of imperfections. Buckling loads are computed for four sets of relevant boundary conditions revealing a strong dependence of the global buckling load on overall end-rotation constraint when local buckling precedes global buckling. A reasonably complete picture emerges, which should be useful for informing decisions on establishing knockdown factors. Experiments are performed using a lateral probe to study the stability landscape for a cylindrical shell with overall end rotation constrained in the first set of tests and then unconstrained in the second set of tests. The nonlinear buckling behavior of spherical shells under external pressure is also examined for both types of imperfections. The buckling behavior of spherical shells is different in a number of important respects from that of the cylindrical shells, particularly regarding the interplay between local and global buckling and the post-buckling load-carrying capacity. These behavioral differences have bearing on efforts to revise buckling design rules. The present study raises questions about the perspicacity of using probe force imperfections as surrogates for geometric dimple imperfections.

Nonlinear Buckling Interaction for Spherical Shells Subject to Pressure and Probing Forces

Elastic spherical shells loaded under uniform pressure are subject to equal and opposite compressive probing forces at their poles to trigger and explore buckling. When the shells support external pressure, buckling is usually axisymmetric; the maximum probing force and the energy barrier the probe must overcome are determined. Applications of the probing forces under two different loading conditions, constant pressure or constant volume, are qualitatively different from one another and fully characterized. The effects of probe forces on both perfect shells and shells with axisymmetric dimple imperfections are studied. When the shells are subject to internal pressure, buckling occurs as a nonaxisymmetric bifurcation from the axisymmetric state in the shape of a mode with multiple circumferential waves concentrated in the vicinity of the probe. Exciting new experiments by others are briefly described.

Discrepancy between boundary conditions and load introduction of full-scale built-in and sub-scale experimental shell structures of space launcher vehicles

Shell buckling experiments are mostly conducted in a displacement controlled manner, that is the displacement at the loaded shell edge is increased and the load applied is measured as reaction force. The corresponding boundary conditions are realized by potting the shell edges. Real shell structures, such as primary structures of space launcher vehicles, are loaded in a load controlled manner and boundary conditions are defined by the adjacent structures and stiffening rings. Within this contribution, the discrepancy between boundary conditions and load introduction of full-scale built-in and sub-scale experimental shell structures of space launcher vehicles is studied numerically. For this purpose, dynamic explicit load controlled buckling analyses were performed using theoretical boundary conditions to idealize built-in conditions in an extreme manner and taking localized perturbations, such as those due to a single perturbation load, geometrical dimple imperfections and circular unreinforced cut outs, into account. The results are compared to displacement controlled shell buckling predictions in which boundary conditions commonly used within shell buckling experiments of sub-scale structures are taken into account.

Determination of realistic worst imperfection for cylindrical shells using surrogate model

Thin-walled cylindrical shells subjected to axial compression are sensitive to initial geometrical imperfections. Firstly, the influence of a single dimple imperfection on the load-carrying capacity of cylindrical shells is investigated by varying the amplitudes, directions and positions of the dimple imperfection. Then a single delta function is introduced to describe the profile of the dimple imperfection in a more general form, which enables to analyze conveniently the effect of the imperfect zone area on the load-carrying capacity of cylindrical shells. Thereafter, a surrogate-based optimization framework of determining the worst realistic imperfection is proposed to study the reduction of the buckling load of cylindrical shells based on a finite number of dimple imperfections. The possible dimple imperfections may be introduced during the service of cylindrical shells. Finally the effectiveness of the present method is demonstrated by an illustrative example.

Collapse strength of longitudinal plate assemblies with dimple imperfections

This study considers the effects of dimple imperfections, i.e. imperfections caused by local accidents, on the ability of a plate assembly to resist compressive loads. Several different parameters for the plate were used, so that a relationship between these and the effect of the dimple imperfection could be developed. One such parameter was the position of the dimple imperfection on the plate. The study was based on finite element calculations. The results obtained were compared with the ones for the same panels without the dimple imperfections. It was found that the effect of the dimple imperfection is higher when positioned near the unloaded edge of the plate. It was also found that the effect on the collapse strength of this type of imperfection depends upon its amplitude and on the slenderness of the plate.

Stability under step loading of infinitely long columns with localized imperfections

The dynamic buckling of a long column with small dimple imperfections resting on a nonlinear foundation and subjected to axial step-loading is studied using a formal multi-variable perturbation expansion. Simple asymptotic formulas are obtained for the dynamic buckling load and lateral deflection in terms of the Fourier transform of the imperfection. It is found that the static and dynamic buckling loads are equal.

Buckling under external pressure of cylindrical shells with dimple shaped initial imperfections

The approximate load-deflection behavior of a simply supported cylindrical shell subjected to external pressure is determined for the case where the shell has a dimple shaped initial deflection. This is accomplished by the use of a “two-timing” perturbation expansion applied to the Kármán-Donnell shell equations. We show that if the shell is imperfection-sensitive to an initial deflection in the shape of the linear buckling mode, then it is also imperfection-sensitive to a dimple imperfection. The degradation in buckling load for the dimple imperfection depends linearly upon the initial deflection amplitude ε whereas for the modal imperfection it is proportional to the two-thirds power of ε.

Effect of a local axisymmetric imperfection on the buckling behaviorof a circular cylindrical shell under axial compression

Local axisymmetric dimple imperfection effects on buckling load of circular cylindrical shell under axial compression