Elastic Postbuckling Research Articles

Parametric control of quasi-zero stiffness mechanisms for vibration isolation at near-zero frequencies

Infra-frequency vibration, including near-zero frequencies, has become a critical limiting factor for developers of next-generation ultra-high precision measurement systems. Mechanisms involving quasi-zero stiffness show potential to solve this new vibration problem. An approach for designing a novel type of the mechanisms for this class is presented. An algorithm is developed and tested for numerical solution of nonlinear elasticity problem in designing the mechanism structural and parametric elements based on FEM models. The elements are spatially prestressed multi-layer designs of polymer composites for modeling elastic postbuckling in large and the effect of a “floating” binder. An algorithm and models following the Nelder-Mead method are used for optimizing the designs. Examples of Euler-beam-shaped composite elements under postbuckling demonstrate a qualitative leap in mechanisms design and efficiency, specifically, in terms of compactness, lightness, and strength. A dramatically expanded control range for the linear section of operating travel with quasi-zero stiffness is obtained. A damping threshold is evaluated, and an optimal type diagram of the mechanism is designed and enumerated as necessary conditions for vibration isolation at near-zero frequencies. The validity and effectiveness of the approach are demonstrated with the results of theoretical design and testing of the prototype for vibration isolation of the ultra-high precision optical system. The learning capability of the mechanisms is estimated by an adapted version of the Levenberg-Marquardt algorithm with stepwise refinement in the test mode using Rosenbrock’s function.

Semi-Analytical Model to Predict the Elastic Post-Buckling Response of Axially Compressed Cylindrical Shells With Tailored Distributed Stiffness

Abstract This study introduces an approximate analytical model to predict the post-buckling response of cylinders with tailored non-uniform distributed stiffness. The shell's wall thickness, and thus its stiffness, is tailored so as to obtain multiple controlled elastic local buckling events when the cylinder is subjected to uniform axial compression. The proposed model treats cylinder segments of different stiffness as individual panels and combines their response by considering them as connected linear or nonlinear springs. The governing equations for the panels are formulated using von Karman's theory and solved by Galerkin's approximate method for a predefined radial deformation. Radial deformation functions are used to improve the model's accuracy, and results show that the model's accuracy increases significantly with the number of considered radial functions. The model's predicted axial response for different cylinders is compared with results from experiments on three-dimensional printed samples. Results indicate that this model accurately predicts the order of the buckling events, while the buckling forces from the model are higher than those measured experimentally.

On the Lateral-Torsional Elastic Post-Buckling and Strength of Channel Steel Beams

This paper aims at shedding new light on the elastic post-buckling behavior and strength (elastoplastic collapse) of steel channel section beams undergoing lateral-torsional (LT) buckling. This new insight is acquired using a two-node geometrically exact beam finite element developed by the author, which can handle large displacements, including torsion-related warping, Wagner effects, plasticity, residual stresses and arbitrary initial configurations (namely geometrical imperfections). Several cross-sections and loading/support conditions are analyzed. The results obtained show that the LT buckling behavior of channels is asymmetric, meaning that the direction of the geometric imperfection can influence significantly the strength of the beams. Moreover, it is demonstrated that the loading and support conditions play a major role, leading to a significant scatter of results which hinder the definition of a single buckling curve for design purposes. Nevertheless, in all cases considered, the buckling strengths obtained fall significantly above those predicted by the current version of Eurocode 3.

Cylindrical Shells with Tunable Postbuckling Features Through Non-Uniform Patterned Thickening Patches

The research reported herein follows the increased interest in buckling-induced functionality for novel materials and devices with a focus on cylindrical shells as a suitable structural prototype. The paper proposes the concept of using patterned thickening patches on the surface of cylindrical shells to modify and control their elastic postbuckling response. Cylindrical shells with non-uniform thickness distributions (NTD) were fabricated through 3D printing to understand rules for pattern designs and then tested under loading-unloading cycles. Strategic thickening patches act as governing imperfections that modify the response type, the number, the location and the sequence of the localized buckling events. The use of patterned thickening patches and their layout provides diverse design opportunities for a desired elastic postbuckling response and can be potentially used in design materials and structures with switchable functionalities.

Controlled elastic postbuckling of bilaterally constrained non-prismatic columns: application to enhanced quasi-static energy harvesters

Axially compressed bilaterally constrained columns, which can attain multiple snap-through buckling events in their elastic postbuckling response, can be used as energy concentrators and mechanical triggers to transform external quasi-static displacement input to local high-rate motions and excite vibration-based piezoelectric transducers for energy harvesting devices. However, the buckling location with highest kinetic energy release along the element, and where piezoelectric oscillators should be optimally placed, cannot be controlled or isolated due to the changing buckling configurations. This paper proposes the concept of stiffness variations along the column to gain control of the buckling location for optimal placement of piezoelectric transducers. Prototyped non-prismatic columns with piece-wise varying thickness were fabricated through 3D printing for experimental characterization and numerical simulations were conducted using the finite element method. A simple theoretical model was also developed based on the stationary potential energy principle for predicting the critical line contact segment that triggers snap-through events and the buckling morphologies as compression proceeds. Results confirm that non-prismatic column designs allow control of the buckling location in the elastic postbuckling regime. Compared to prismatic columns, non-prismatic designs can attain a concentrated kinetic energy release spot and a higher number of snap-buckling mode transitions under the same global strain. The direct relation between the column’s dynamic response and the output voltage from piezoelectric oscillator transducers allows the tailorable postbuckling response of non-prismatic columns to be used as multi-stable energy concentrators with enhanced performance in micro-energy harvesters.

Harnessing Seeded Geometric Imperfection to Design Cylindrical Shells With Tunable Elastic Postbuckling Behavior

Geometric imperfection, known as a detrimental effect on the buckling load of cylindrical shells, has a new role under the emerging trend of using buckling for smart purposes. Eigenshape-based geometries were designed on the shell surface with the aim of tailoring the postbuckling response. Fourteen seeded geometric imperfection (SGI) cylinders were fabricated using polymer-based 3D printing, and their postbuckling responses were numerically simulated with a general-purpose finite element program. Results on the prototyped SGI cylinders showed a tunable elastic postbuckling response in terms of initial and final stiffness, the maximum load drop from mode switching, and the number of snap-buckling events. A response contour and discrete map is presented to show how the number of waves in the axial and circumferential directions in the seeded eigenshape imperfection can control the elastic postbuckling response. SGI cylinders provide diverse design opportunities for controllable unstable response and are good candidates for use in smart and adaptive materials/structures.

Elastic postbuckling response of axially-loaded cylindrical shells with seeded geometric imperfection design

Elastic instabilities (such as buckling) have been recognized as a promising phenomenon to design smart materials and mechanical systems. Thin-walled cylindrical shells under axial compression can attain multiple bifurcation points in their postbuckling regime due to the natural transverse deformation restraint provided by their geometry; but harnessing such behavior for smart purposes is lacking extensive study due to its notoriously high imperfection sensitivity. In this paper, the concept of seeded geometric imperfection (SGI) design is proposed to modify and control the elastic postbuckling behavior of cylindrical shells. Eigenvalue-based mode shapes were used as basic geometric forms to generate a seeded imperfection. Prototyped SGI cylindrical shells were fabricated through 3D printing and tested under loading–unloading cycles. Numerical and experimental results suggest that the SGI cylindrical shells are less sensitive to initial imperfections and load variation than uniform ones. Cylindrical shells with seeded geometry can be potentially used in the design of smart devices and mechanical systems such as energy harvesters and self-powered sensors.

Elastic Buckling and Post-Buckling of Von Mises Planar Trusses

The article deals with the computational model of an elastic von Mises planar truss. The description of a mathematical concept, which is intended for the creation of computational programmes based on a finite number of segments, is presented. The mathematical solution is suitable for the analysis of load-deflection curves. Structural deformation is evaluated by seeking the minimal potential energy. The article examines the effects of change in the vertical displacement of the top joint on strut axes. The step by step incremental method is used in combination with the Newton-Raphson method. The presented study is aimed at the evaluation of the force in the bifurcation point, which determines the moment when loading of the model causes passing from the pre-critical effect (attainment of maximum vertical load action) to the post-critical effect. Symmetric and asymmetric initial axis imperfections are considered and relevant symmetric and asymmetric shapes of buckling are identified. The stability problem of the von Mises truss is discussed in connection with the random effects of imperfections.

Tailoring the elastic postbuckling response of cylindrical shells: A route for exploiting instabilities in materials and mechanical systems

Postbuckling response, long considered mainly as a failure limit state is gaining increased interest for smart applications, such as energy harvesting, frequency tuning, sensing, actuation, etc. This letter explores the potential of cylindrical shells under axial compression, for which localized buckling events along with high-rate local deformations can be attained in their elastic postbuckling regime under smaller axial system deformation input, as a prototype for harnessing elastic instabilities. Three avenues are presented to tailor and control the elastic postbuckling response of axially-compressed shells: (1) by introducing seeded geometric imperfections (SGI); (2) by introducing non-uniform stiffness distributions (NSD); and (3) by providing lateral constraints and interactions (LCI). Prototyped cylindrical shells were fabricated through 3D printing and tested under loading–unloading cycles. Experimental results show that, with appropriate selection of geometry, material, and stiffness distribution, these three concepts offer significant advantages over uniform cylindrical shells for use of their notoriously unreliable elastic postbuckling response. This work provides new knowledge on the possibilities and means to design the cylindrical shells with controlled elastic postbuckling behavior and opens new avenues for using this structural form for applications in smart materials and structures.

Self-Powered Piezo-Floating-Gate Smart-Gauges Based on Quasi-Static Mechanical Energy Concentrators and Triggers

Changes in physical processes like ambient temperature or pressure variations occur at frequencies that are significantly lower than 1 Hz. This poses a challenge for designing self-powered sensors that monitor these quasi-static physical processes and at the same time scavenge operational energy for sensing, computation, and storage from the signal being monitored. In this paper, we present a novel paradigm for designing a self-powered sensor/data logger that exploits the physics of negative-stiffness mechanical energy concentrators with the physics of our previously reported piezoelectricity driven impact ionized hot-electron injection (p-IHEI)-based sensors. The operational principle is based on the sudden transitions from unstable mode branch switching during the elastic postbuckling response of slender columns, which are used to generate high-frequency deformations as an input to the p-IHEI-based sensor. The experimental results demonstrate that the proposed self-powered sensor based on an integrated circuit fabricated in a 0.5-μm CMOS technology can count and record the number of quasi-static input events with frequencies spanning less than 1 Hz.

Tailoring the elastic postbuckling response of thin-walled cylindrical composite shells under axial compression

The buckling of cylindrical shells has long been regarded as an undesirable phenomenon, but increasing interests on the development of active and controllable structures open new opportunities to utilize such unstable behavior. In this paper, approaches for modifying and controlling the elastic response of axially compressed laminated composite cylindrical shells in the far postbuckling regime are presented and evaluated. Three methods are explored (1) varying ply orientation and laminate stacking sequence; (2) introducing patterned material stiffness distributions; and (3) providing internal lateral constraints. Experimental data and numerical results show that the static and kinematic response of unstable mode branch switching during postbuckling response can be modified and potentially tailored.

Thermomechanical Elastic Post-Buckling of Functionally Graded Materials Plate with Random System Properties

This paper presents the stochastic post-buckling response of elastically supported FGM plate with random system properties subjected to uniform and nonuniform temperature change with temperature-dependent and -independent material properties. The FGMs plate is supported with two parameters of Pasternak foundation with Winkler cubic nonlinearity. The basic formulation is based on higher-order shear deformation theory (HSDT) with von-Karman nonlinearity using modified C0 continuity. A direct iterative-based nonlinear finite element method combined with first-order perturbation technique is used to compute the second-order statistics (mean and coefficient of variation) of post-buckling response of FGM plates.

Amos Henry Chilver FREng, Baron Chilver of Cranfield. 30 October 1926 — 8 July 2012

Henry Chilver was a structural engineer and educationalist. During his early career at Cambridge and University College London, he and his team of researchers made significant contributions, both theoretically and experimentally, to the understanding of imperfection sensitivity in the elastic post-buckling of thin-walled structural components. Awareness of this sensitivity was essential for the safe design of load-carrying shells that were increasingly being used in nuclear reactors and aerospace structures. As Vice-Chancellor of Cranfield Institute of Technology he focused on the application of knowledge rather than merely on its acquisition and led the way in harnessing research relevant to the demands of industry. With his incisive intellect, innovative thinking and farsighted vision he was also in great demand as an adviser to the government, overseeing advisory committees, and holding chairmanships and directorships of numerous companies.

Modified Elastofiber Element for Steel Slender Column and Brace Modeling

An efficient beam element, the modified elastofiber (MEF) element, has been developed to capture the overall features of the elastic and inelastic responses of slender columns and braces under axial cyclic loading without unduly heavy discretization. It consists of three fiber segments, two at the member ends and one at midspan, with two elastic segments sandwiched in between. The segments are demarcated by two exterior nodes and four interior nodes. The fiber segments are divided into 20 fibers in the cross section that run the length of the segment. The fibers exhibit nonlinear axial stress-strain behavior akin to that observed in a standard tension test of a rod in the laboratory, with a linear elastic portion, a yield plateau, and a strain-hardening portion consisting of a segment of an ellipse. All the control points on the stress-strain law are user defined. The elastic buckling of a member is tracked by updating both exterior and interior nodal coordinates at each iteration of a time step and checking force equilibrium in the updated configuration. Inelastic postbuckling response is captured by fiber yielding, fracturing, and/or rupturing in the nonlinear segments. The key features of the element include the ability to model each member using a single element, easy incorporation of geometric imperfection, partial fixity support conditions, member susceptibility to fracture defined in a probabilistic manner, and fiber rupture leading to complete severing of the member. The element is calibrated to accurately predict the Euler critical buckling load of box and I sections with a wide range of slenderness ratios (L/r=40, 80, 120, 160, and 200) and support conditions (pinned-pinned, pinned-fixed, and fixed-fixed). Elastic postbuckling of the Koiter-Roorda L frame (tubes and I sections) with various member slenderness ratios (L/r=40, 80, 120, 160, and 200) is simulated and shown to compare well against second-order analytical approximations to the solution even when using a single-MEF element to model each leg of the frame. The inelastic behavior of struts under cyclic loading observed in the experiments of Black et al., Fell et al., and Tremblay et al. is accurately captured by single-MEF-element models. A FRAME3D model (using MEF elements for braces) of a full-scale six-story braced frame structure that was pseudodynamically tested at the Building Research Institute of Japan subjected to the 1978 Miyagi-Ken-Oki earthquake record is analyzed and shown to closely mimic the experimentally observed behavior.

Elastic postbuckling stiffness of biaxially compressed rectangular plates

The elastic buckling and postbuckling response of biaxially compressed plates is analysed with main focus on change of in-plane postbuckling stiffness under prescribed plate shortenings rather than the more normal load control. An analytical closed-form single-degree-of-freedom solution is derived and compared with an analytical multi-degree-of-freedom model. Results for the analytical multi-degree-of-freedom-model are obtained by solving numerically the equilibrium equations using the perturbation expansion technique with arc length control in an incremental scheme. Both models are applied to specific plate examples and comparisons with results obtained using a commercial nonlinear finite element program are included.

Variable order secant matrix technique applied to the elastic postbuckling of arches

AbstractThis paper presents the implementation of Reissner's (J. Appl. Math. Phys. 1972; 23:795–804) strain–displacement relations for a planar beam using a variable order secant matrix (VOSM) technique. The secant matrices are formed using a Taylor expansion of the strain energy and hence the order of the secant and tangent matrices to be included in the nonlinear analysis can be varied depending upon the degree of nonlinearity involved in the problem. Nevertheless, a maximum of 24 matrices is sufficient to obtain predicted internal stress resultants to within 0.2% of the converged solution in all the problems tested. Using a three‐noded isoparametric beam element, the accuracy of the VOSM method is demonstrated for the elastic postbuckling of arches. Copyright © 2007 John Wiley & Sons, Ltd.

Design and mechanical properties of insect cuticle

Since nearly all adult insects fly, the cuticle has to provide a very efficient and lightweight skeleton. Information is available about the mechanical properties of cuticle—Young's modulus of resilin is about 1 MPa, of soft cuticles about 1 kPa to 50 MPa, of sclerotised cuticles 1–20 GPa; Vicker's Hardness of sclerotised cuticle ranges between 25 and 80 kgf mm −2; density is 1–1.3 kg m −3—and one of its components, chitin nanofibres, the Young's modulus of which is more than 150 GPa. Experiments based on fracture mechanics have not been performed although the layered structure probably provides some toughening. The structural performance of wings and legs has been measured, but our understanding of the importance of buckling is lacking: it can stiffen the structure (by elastic postbuckling in wings, for example) or be a failure mode. We know nothing of fatigue properties (yet, for instance, the insect wing must undergo millions of cycles, flexing or buckling on each cycle). The remarkable mechanical performance and efficiency of cuticle can be analysed and compared with those of other materials using material property charts and material indices. Presented in this paper are four: Young's modulus—density (stiffness per unit weight), specific Young's modulus—specific strength (elastic hinges, elastic energy storage per unit weight), toughness—Young's modulus (fracture resistance under various loading conditions), and hardness (wear resistance). In conjunction with a structural analysis of cuticle these charts help to understand the relevance of microstructure (fibre orientation effects in tendons, joints and sense organs, for example) and shape (including surface structure) of this fibrous composite for a given function. With modern techniques for analysis of structure and material, and emphasis on nanocomposites and self-assembly, insect cuticle should be the archetype for composites at all levels of scale.

Elasto-plastic buckling and postbuckling of arches subjected to a central load

This paper presented a rational finite curved beam-element model for 3D nonlinear elasto-plastic analysis of arches. The finite element model incorporates the effects of large twist rotations of the cross-section and has consistent sampling scheme of sampling points over the cross-section and so it can perform the elastic and elasto-plastic flexural–torsional buckling and postbuckling analysis of arches in combined bending and compression actions. Elastic and elasto-plastic flexural–torsional buckling and postbuckling behaviour of arches that are subjected to a central concentrated load has been investigated using the rational finite element model. It has been found that the slenderness, included angle, and the torsional parameter of an arch play important roles in the elastic and elasto-plastic buckling. In addition, the number of the inflexion points is important for the elastic and elasto-plastic buckling of fixed arch. Yielding is a significant factor for the elasto-plastic buckling and postbuckling behaviour. When the included angle of a stocky arch is not large, its elasto-plastic buckling load is much lower than its elastic buckling load. When the included angle a stocky arch is large, the end support condition plays an important role in the buckling and postbuckling behaviour of the arch. The elasto-plastic buckling load of a stocky pin-ended arch with a large included angle is equal to its elastic counterpart. The elasto-plastic buckling load of a stocky fixed arches with a large included angle is lower than its elastic counterpart. For a slender arch, its elasto-plastic buckling load is also equal to its elastic buckling load. In general, the elasto-plastic postbuckling load carrying capacity of an arch decreases when the arch continues to deform while its elastic postbuckling load carrying capacity can increase. For slender shallow arches, the elastic postbuckling response is stiffer because of the relaxation of axial load and moment distribution.

Elastic postbuckling with nonlinear constraints

Koiter’s asymptotic theory of initial postbuckling and imperfection sensitivity of elastic structures is expanded such that almost any auxiliary condition can be handled directly. The theory, which employs Lagrange multiplier techniques, is valid for strain measures that are quadratic in the displacements, i.e. it is exact for Lagrangian strain measures, and auxiliary conditions and loadings that are quartic in their arguments. Although the general formulas are rather lengthy and complicated in the general case they simplify considerably in almost all cases of practical interest. Frequently the formulas are only slightly more complex than in the equivalent situation without auxiliary conditions.The method is applied to the example of an elastic inextensible circular ring but is suitable for all elastic structures which comply with the above mentioned requirements regarding strain measure and auxiliary conditions.

Axisymmetric elastic postbuckling of circular plates

This paper deals with the elastic postbuckling behaviour of axisymmetric thin plates. The analysis is based on an alternative boundary integral equation formulation which has been applied successfully in connection with the large deflection analysis of plates and shells. Numerical results are obtained and compared with corresponding ones available in the literature. Further applications of the proposed formulation to more advanced problems are also discussed.