Loading Imperfections Research Articles

Experimental and numerical studies on the elasto-plastic buckling response of cylindrical shells with spigot support under axial compression

In this work, an experimental and numerical methodology is proposed to study the elasto-plastic buckling response of thin cylindrical shells with spigot support under axial compression. A multi-3D digital image correlation system is used to measure the geometrical imperfections as well as the buckling response of the shell. Further, a coordinate measuring machine is employed to measure the edge loading imperfections of the shell. Next, interpolation methods are employed to transfer the measured imperfections to a finite element (FE) model. The effect of interference between the spigot and shell on the nonlinear buckling response of the shell is studied. The numerical results of FE models embedded with the measured edge and radial imperfections are compared with the experimental response. This study emphasizes the importance of including geometrical/loading imperfections for accurately estimating the buckling load of cylindrical shells used in launch vehicle applications.

Manufacture and Buckling Test of a Variable-Stiffness, Variable-Thickness Composite Cylinder Under Axial Compression

Variable-angle tow (VAT) manufacturing methods significantly increase the design space for elastic tailoring of composite structures by smoothly changing fiber angle and ply thickness across a component. Rapid tow shearing (RTS) is a VAT manufacturing technique that uses in-plane shearing (rather than in-plane bending) to steer tows of dry or pre-impregnated fibers. RTS offers a number of benefits over conventional bending-driven steering processes, including tessellation of adjacent tow courses; no overlaps or gaps between tows; and no fiber wrinkling or bridging. Further to this, RTS offers an additional design variable: fiber orientation to tow thickness coupling due to the volumetric relation between tow shearing and the tow’s thickness and width. Previous computational work has shown that through a judicious choice of curvilinear fiber trajectories along a cylinder’s length and across its circumference, the imperfection sensitivity of cylindrical shells under axial compression can be reduced and load-carrying capacity increased. The present work aims to verify these predictions by manufacturing and testing two cylinders: an RTS cylinder and a straight-fiber, quasi-isotropic cylinder as a benchmark. The tow-steered manufacturing process, imperfection measurements, instrumentation, and buckling tests of both cylinders are discussed herein. The experimental tests results are compared against high-fidelity geometrically nonlinear finite element models that include measured geometric and loading imperfections before and during the tests. Finally, a discussion is provided on the outstanding challenges in designing and manufacturing RTS cylinders for primary aerostructures.

Analysis and validation of a scaled, launch-vehicle-like composite cylinder under axial compression

Launch vehicle structures, such as payload adapters and interstages, are increasingly designed and constructed using composite materials due to their high stiffness- and strength-to-weight ratios. Therefore, it is important to develop a validated finite element modeling methodology for designing and analyzing composite launch-vehicle shell structures. This can be achieved, in part, by correlating high-fidelity numerical models with test data. Buckling is often an important failure mode for cylindrical shells, and the buckling response of such structures is also often quite sensitive to imperfections in geometry and loading. Hence, it is crucial to understand the model parameters and details required to accurately predict the buckling load and behavior of composite cylindrical shells, especially if the shell is buckling critical. The inclusion of as-built features, such as radial imperfections, thickness variations, and loading imperfections can help improve the correlation between test and analysis. To demonstrate such an approach, a validated modeling methodology that was used to predict the buckling behavior of a scaled component for a launch-vehicle-like structure is presented, and results from the model are compared with experimental results. The modeling approach presented herein was used to successfully predict the buckling behavior.

Buckling of thick elasto-visco-plastic egg shells under external pressure: Experiments and bifurcation analysis

The buckling of rate dependent structures is not a topic often discussed in the literature. Methods to predict the buckling of such structures exist, but only few experimental works have been published, and even less when considering thick shell structures. In this work we are interested in the buckling of thick elasto-visco-plastic hemi-egg shells. We follow a modelling/experimental approach to analyse such problem. Buckling experiments at room temperature on hemi-egg shells subjected to external pressure are performed. 3D digital image correlation is used to measure the displacement fields on the inner surface of the hemi-egg shell during the buckling experiments. Due to the specimen shape and their loading, they experience non-proportional load paths. The effect of a non-proportional load path on the buckling behaviour has already been discussed many times in the literature for elasto-plastic materials. This work also intends to assess this point for a rate dependent material. A buckling prediction model is used to predict the buckling of hemi-egg shells. This model couples Bodner’s approach with the deformation theory. It is used to estimate the buckling critical times and the buckling modes based on a preliminary finite element analysis. The effects of geometrical and loading imperfections on the buckling behaviour of the hemi-egg shells are also discussed. Finally, the experimental results are compared to the buckling predictions. A good correlation is observed between numerical results and experiments.

Numerical study and experiments on the elasto-visco-plastic bifurcation buckling of thick anisotropic plates

The present work investigates the elasto-visco-plastic buckling of thick plate structures. Methods to estimate the buckling of thin elasto-visco-plastic shells and plates are proposed in literature. To the knowledge of the authors, only few works present experimental results, and none treat the experimental elasto-visco-plastic buckling of thick plate. Thick rectangular plates are chosen as a structure of interest to study elasto-visco-plastic buckling. A modelling/experimental approach is followed to solve such problem. Buckling experiments at room temperature on thick elasto-visco-plastic plates subjected to compressive load are performed. 3D digital image correlation (3D-DIC) is used to measure displacement fields on plate surface during buckling experiments. The experiments are also modelled through a finite element model. Bodner’s approach is coupled to Hencky’s deformation theory to predict buckling of elasto-visco-plastic plates. Bifurcation analysis are performed to estimate buckling critical stresses and strains, and buckling modes. Geometrical and loading imperfections are also discussed. A good correlation is observed between numerical results and experiments. The limitation of Bodner’s approach is also discussed.

A parametric study on out-of-plane instability of doubly reinforced structural walls. Part II: Experimental investigation

The effects of different parameters on the evolution of Out-Of-Plane (OOP) instability in rectangular walls were studied using a validated FEM model. Based on this parametric analysis, a test matrix was designed to experimentally investigate the progression of this mode of failure in rectangular walls. Results of the parametric study were presented within a companion paper, while the experimental observations as well as the numerical versus experimental comparisons are presented in this paper. The wall specimens were tested under in-plane cyclic loading and they exhibited different OOP responses. The development of OOP mechanisms was observed in all tested specimens, albeit with different values of maximum OOP displacement and followed by different types of instability. Progression of OOP deformation was suppressed by other failure modes in all specimens except one which failed due to large OOP displacement and global instability. In this study, the progression of OOP deformation and its relationship with the inelastic strain gradients of the longitudinal reinforcement along the length and height of the specimens are scrutinized and the numerical predictions are compared with the test measurements. The experimental findings regarding the evolution of this mode of failure and the influential parameters are in good agreement with the numerical predictions. However, the onset of OOP deformation typically had a delay in the numerical model when compared to the test results. One of the factors inducing this difference could be the effect of inherent and mostly unmeasurable eccentricities, including material, geometry as well as loading imperfections, which were not included in the model.

Buckling of cylindrical shells under axial compression with loading imperfections: An experimental and numerical campaign on low knockdown factors

Abstract Thin-walled cylindrical shells are primary structures in aerospace, marine and civil engineering. A major loading scenario for these imperfection sensitive shells is axial compression. For this load case, there is a critical disagreement between the theoretical and experimental critical load. This difference is mainly contributed due to shape deviations of the shell middle plane which are commonly described as geometric imperfections. However, some deviations between theoretical and experimental buckling loads are sometimes so severe that other imperfection types are possibly to blame. This article describes experimental and numerical studies on the influence of loading imperfections on the buckling load of thin-walled cylinders. A global loading imperfection was applied by mistake during a buckling test of a thin-walled cylinder which led to a severe buckling load reduction and the corresponding load level was similar to the post-buckling load. Also, a series of experimental studies with localized loading imperfections is described which significantly reduced the buckling load of CFRP cylinders. The experimental results of this article represent an example for some of the very low buckling knockdown factors from early experimental campaigns and give insights in how to avoid these critical imperfections in experimental buckling campaigns.

Physics-Informed Artificial Neural Network Approach for Axial Compression Buckling Analysis of Thin-Walled Cylinder

Unstable buckling leads to the failure of thin-walled cylinders under axial compression. Because of the manufacturing and loading imperfections, a significant variation was observed between experimental and theoretical buckling load. The current design criteria use a knockdown factor to estimate the buckling load of thin-walled structures. However, the design criteria employed a very conservative knockdown factor, because it is difficult to be accurately predicted. To improve the certainty of the knockdown factor, in this paper, a physics-informed artificial neural network (PANN) was employed to predict the thin-walled cylinder buckling load using experimental data collected by Seide. PANN has the potential to improve prediction and enforce some physical laws. A beam problem was presented to demonstrate the PANN capability. Then, the cylinder buckling analysis was carried out with PANN. An inequality constraint was applied to force the neural-network-predicted buckling load to be equal or smaller than the corresponding experimental data. The result shows that PANN can reach excellent prediction accuracy and is able to predict the buckling load equal to or smaller than the corresponding experimental buckling load. Additionally, both ANN and PANN are able to take the unique influential parameters as inputs and do not have presumed expression and thus have the potential to yield a less conservative prediction than the statistical equation.

Nonlinear stability analysis of FGM shallow arches under an arbitrary concentrated radial force

The nonlinear in-plane buckling behaviour of pinned-pinned shallow circular arches made of functionally graded material (FGM) is investigated using a one-dimensional Euler–Bernoulli model. The effect of the bending moment on the membrane strain is included. The material properties vary along the thickness of the arch. The external load is a radial concentrated force at an arbitrary position. The pre-buckling and buckled equilibrium equations are derived using the principle of virtual work. Analytical solutions are given for both bifurcation and limit point buckling. Comprehensive studies are performed to find the effect of various parameters on the buckling load and on the behaviour. The major findings: (1) arches have multiple stable and unstable equilibria; (2) when the load is at the crown, the lowest buckling load is related to bifurcation buckling for most geometries and material compositions but when the external force is replaced, only limit point buckling is possible; (3) the position of the load has huge influence on the buckling load; (4) such arches are sensitive to small loading imperfections when loaded in the vicinity of the crown. The paper intends to improve and extend the existing knowledge about the behaviour of pinned-pinned shallow FGM arches under arbitrary concentrated radial load.

Transverse vibrations induced by longitudinal excitation in beams with geometrical and loading imperfections

With structural health monitoring techniques based on measuring transverse vibrations of beam-like components it can be difficult to provide transverse excitation, while longitudinal excitation can be easier. We experimentally investigate how transverse vibrations in a beam can be excited by a longitudinal hammer impact. We carry out a comparative study on the measured transverse natural frequencies and frequency response coherence for different input and output locations and directions. It is shown that transverse vibrations are excited regardless of the impact locations and directions. Theoretical explanation to this counter-intuitive phenomenon is provided in terms of various imperfections associated with beams and impacts.

Probabilistic perturbation load approach for designing axially compressed cylindrical shells

The buckling load of cylindrical shells is heavily dependent on geometric imperfections and other non-traditional imperfections such as scattering wall thicknesses or loading imperfections. In this paper, the probabilistic perturbation load approach (PPLA) is proposed. This procedure is independent from costly measurements of geometric imperfections and is able to include various kinds of non-traditional imperfections by integrating the single perturbation load approach into a semi-analytical probabilistic framework. It is shown that the PPLA leads to design loads which are lower than the experimental buckling loads and much less conservative than the ones obtained from NASA SP-8007 in all investigated cases.

The use of initial imperfection approach in design process and buckling failure evaluation of axially compressed composite cylindrical shells

Thin-walled cylindrical shells are susceptible to buckling failures caused by the axial compressive loading. During the design process or the buckling failure evaluation of axially-compressed cylindrical shells, initial geometric and loading imperfections are of important parameters for the analyses. Therefore, the engineers/designers are expected to well understand the physical behaviours of shell buckling to prevent unexpected serious failure in structures. In particular, it is widely reported that no efficient guidelines for modelling imperfections in composite structures are available. Knowledge obtained from the relevant works is open for updates and highly sought. In this work, we study the influence of imperfections on the critical buckling of axially compressed cylindrical shells for different geometries and composite materials (Glass Fibre Reinforced Polymer (GFRP), Carbon Fibre Reinforced Polymer (CFRP)) and aluminium using the finite element (FE) analysis. Two different imperfection techniques called eigenmode-affine method and single perturbation load approach (SPLA) were adopted. Validations of the present results with the published experimental data were presented. The use of the SPLA for introducing an imperfection in axially compressed composite cylindrical shells seemed to be desirable in a preliminary design process and an investigation of a buckling failure. The knockdown factors produced by the SPLA were becoming attractive to account for uncertainties in the structure.

Stochastic analysis of imperfection sensitive unstiffened composite cylinders using realistic imperfection models

The important role of imperfections on decreasing the buckling load of structural cylinders has been investigated by scientists and engineers for the past century, yet there is currently no method that is able to stochastically replicate the full range of realistic imperfections for a full account of possible buckling loads. This drawback impairs optimised design as designers are restrained to using an outdated and conservative design philosophy which dates from 1968. Modern manufacturing methods and materials such as composites require new, optimised design measures to take full advantage of their efficiencies. Stochastic analyses can optimise and improve the reliability of such cylinders through accurate prediction of the range of conceivable buckling loads by realistic simulation and sensitivity analyses. A stochastic procedure which realistically models imperfection sensitive composite shells is investigated in this paper. Monte-Carlo simulations of axially compressed cylinders with the full range of imperfection types are performed to show that the stochastic methods described here are able to accurately capture the scatter in the buckling load introduced from the imperfections. The results from a sensitivity analysis indicate that loading imperfections play the largest role in reducing the buckling load knockdown factors of the shell.

Dynamic buckling of partially-sway frames with varying stiffness using catastrophe theory

This work deals with the static and dynamic stability analysis of imperfect partially-sway frames with non-uniform columns. The examined two-bar frames are elastically supported and subjected to an eccentrically vertical load at their joint. Through a linear stability analysis, the effect of the taper ratio of the column cross-section on the buckling capacity of the partially-sway frame is thoroughly discussed. Using a non-linear method an accurate formula has been established for determining the exact asymmetric bifurcation point associated with the maximum load carrying capacity. These findings have been re-derived more readily using Catastrophe Theory (CT) and considering the frame as a one degree-of-freedom (1-DOF) system through an efficient technique. A local analysis allows us to classify, after reduction, the total potential energy (TPE) function of the system to one of the seven elementary Thom׳s catastrophes (with known properties) and to obtain static and dynamic singularity as well as bifurcational sets. It has been found that geometrical and loading imperfections, which are always present in structural engineering problems, have a significant effect on the dynamic buckling loads. The efficiency of the present approach is illustrated via several examples, while results from finite element analyses are in good agreement with the analytical solution presented herein.

Stability of Steel Columns with Non-Uniform Cross-Sections

In this work, non-uniform steel members with or without initial geometrical or loading imperfections, that are loaded by axial forces applied concentrically or eccentrically and by concentrated moments applied at the ends or at intermediate points, are studied. More specifically, steel members with varying cross-sections, tapered or stepped or members consisting by two different tapered parts are considered. The formulation presented in this work is based on solving the governing equation of the problem through a numerical method where the eigenshapes of the member are employed. A failure plasticity criterion is introduced for members especially the short ones that will never reach the elastic critical buckling load. Although only the simply supported beam-column case is studied herein, it is obvious that the method can be extended to multi-span beams and frames, by employing the corresponding eigenshapes. Useful diagrams are presented for both the critical buckling loads and the equilibrium paths showing the influence of the main characteristics of the beam-column.

PROBABILISTIC DESIGN OF AXIALLY COMPRESSED COMPOSITE CYLINDERS WITH GEOMETRIC AND LOADING IMPERFECTIONS

The discrepancy between the analytically determined buckling load of perfect cylindrical shells and experimental test results is traced back to imperfections. The most frequently used guideline for design of cylindrical shells, NASA SP-8007, proposes a deterministic calculation of a knockdown factor with respect to the ratio of radius and wall thickness, which turned out to be very conservative in numerous cases and which is not intended for composite shells. In order to determine a lower bound for the buckling load of an arbitrary type of shell, probabilistic design methods have been developed. Measured imperfection patterns are described using double Fourier series, whereas the Fourier coefficients characterize the scattering of geometry. In this paper, probabilistic analyses of buckling loads are performed regarding Fourier coefficients as random variables. A nonlinear finite element model is used to determine buckling loads, and Monte Carlo simulations are executed. The probabilistic approach is used for a set of six similarly manufactured composite shells. The results indicate that not only geometric but also nontraditional imperfections like loading imperfections have to be considered for obtaining a reliable lower limit of the buckling load. Finally, further Monte Carlo simulations are executed including traditional as well as loading imperfections, and lower bounds of buckling loads are obtained, which are less conservative than NASA SP-8007.

Stability and initial post-buckling behaviour of frame system with vertical bracings

Steel halls are assembled of plane thin-walled frames stiffened with vertical bracings consisting of two crossed rods in order to fulfil requirements concerning stability and stiffness of horizontal loads. In the paper, using stationary conditions of the total potential energy, a stability analysis and initial post-buckling behaviour of the construction are investigated. The hall construction is modelled by rigid plane frames, elastically connected on upper ends with the roof and in one or more spans with vertical bracings. It has been proved that the obtained bifurcation point is unsymmetrical and unstable, indicating that a reduction of the critical loads is possible due to some geometrical and loading imperfections.

Nonlinear buckling analysis of simple geometrically imperfect frames

In this paper, the effect of geometrical imperfections due to joint angle deviations of a simple rectangular frame is thoroughly discussed. It has been found that the deviation from the right angle of the joint with respect to the center-lines of the two members of the frame affects significantly the buckling strength of the frame. It is also shown that such geometrical imperfections do not always decrease the strength of such structures but sometimes, even though loading imperfections exist, they act on the favorable side regarding its strength. The influence of various geometric parameters such as slenderness ratio as well as length and moment of inertia ratio of the members on the critical load of the imperfect frame is thoroughly discussed and representative case studies are presented.

Design Analysis for Buckling of Tanks and Silos

When subject to combinations of internal pressure and axial compression, experience shows that the often thin cylindrical shell walls of tanks and silos are prone to a local form of buckling failure that is essentially axisymmetric and is strongly influenced by the nature of the end or intermediate support conditions. This paper will outline the basis of an analytical solution to the inherently nonlinear elastic-plastic buckling into axisymmetric modes, in which end constraints and arbitrary radial pressure loading are modeled as essentially loading imperfections on an otherwise linear, classical, critical buckling formulation. Closed form analytical expressions for the imperfect behavior, including geometric imperfections, are then used to predict either first surface or full section plasticity, allowing buckling failure to be summarized in a form that is closely analogous to the Ayrton-Perry formula for the buckling of columns. It is suggested that a suitably simplified form of this general approach to axisymmetric buckling, when combined with similarly simple methods for the buckling into nonaxisymmetric modes, could provide an important alternative to the current procedures for the buckling design of thin-walled, cylindrical, tanks and silos.

A Motion Amplifier Using an Axially Driven Buckling Beam: II. Modeling and Analysis

In this paper, the nonlinear behavior of a motion amplifier used to obtain large rotations from small linear displacements produced by a piezoelectric stack is studied. The motion amplifier uses elastic (buckling) and dynamic instabilities of an axially driven buckling beam. Since the amplifier is driving a large rotary inertia at the pinned end and the operational frequency is low compared to the resonant frequencies of the beam, the mass of the buckling beam and the dynamics of the PZT stack are neglected and the system is modeled as a single-degree-of-freedom, nonlinear system. The beam simply behaves as a nonlinear rotational spring having a prescribed displacement on the input end and a moment produced by the inertial mass acting on the output end. The moment applied to the mass is then a function of the beam end displacement and the mass rotation. The system can, thus, be modeled simply as a base-excited, spring–mass oscillator. Results of the response for an ideal beam using this reduced-order model agree with the experimental data to a high degree. Inclusion of loading and geometric imperfections show that the response is not particularly sensitive to these imperfections. Parameter studies for the ideal buckling beam amplifier were conducted to provide guidance for improving the design of the motion amplifier and finding the optimal operating conditions for different applications.