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The quest for safe lower bounds to the elastic buckling of axially loaded circular cylindrical shells has exercised researchers for the past 100 years. Recent work bringing together the capabilities of nonlinear numerical simulation, interpreted within the context of extended linear classical theory, has come close to achieving this goal of defining safe lower bounds. This paper briefly summarizes some of the important predictions emerging from previous work and presents new simulation results that confirm these earlier predictions. In particular, we show that for a specified maximum amplitude of the most sensitive, eigenmode-based geometric imperfections, normalized with respect to the shell thickness, lower bounds to the buckling loads remain constant beyond a well-defined value of the Batdorf parameter. Furthermore, we demonstrate how this convenient means of presenting the imperfection-sensitive buckling loads can be reinterpreted to develop practical design curves which provide safe, but not overly conservative, design loads for monocoque cylinders with a given maximum permitted tolerance of geometric imperfection. Hence, once the allowable manufacturing tolerance is specified during design or is measured post-manufacturing, the greatest expected knockdown factor for a shell of any geometry is defined. With the recent research interest in localized imperfections, we also attempt to reconcile their relation to the more classical, periodic, and eigenmode-based imperfections. Overall, this paper provides analytical and computational arguments that motivate a shift in focus in defect-tolerant design of thin-walled cylinders--away from the knockdown experienced for a specific geometric imperfection and towards the worst possible knockdown expected for a specified manufacturing tolerance. [ABSTRACT FROM AUTHOR]

The present work discusses buckling and vibration characteristics of axially functionally graded (AFG) graphene platelet (GPL) composite beams exposed to axially varying loads (AVLs). Timoshenko beam composition with five different types of axial grading GPLs subjected to six different types of AVLs are studied. The effective elastic properties are obtained using Halpin-Tsai model and the equations of motion are obtained following the Hamilton's principle. Then the equations are solved for buckling and vibration analysis using the Ritz method. Influences of nature of axial grading of GPLs and load, content of GPL, and structural boundary conditions are investigated through detailed parametric studies. It is found that the grading pattern of GPLs not only influences the buckling load, but also changes buckling mode shapes of the beam at specific type of AVL. Furthermore, results reveal that buckling and vibration characteristics of beam enhanced in case of AFGM-A type for most of the load cases studied. The proposed study will be helpful for the structural engineers to select the nature of graded distribution of GPLs for the given type of AVL and design the structural member. [ABSTRACT FROM AUTHOR]

This paper investigates the size-dependent vibration and buckling behavior of nanoplates described by the nonlocal Kirchhoff model analytically. The nonlocal and local constitutive relations, governing equations and boundary conditions are comprehensively discussed. A shifted governing equation in terms of local stress resultants with associated local boundary conditions is chosen in the current work. The direct separation of variables method is formulated based on the Hamiltonian dual form of the governing differential equation to address the eigenvalue problems of nonlocal rectangular plates with each edge either simply supported or clamped. The closed-form eigen-solutions are acquired for the first time for the cases where two adjacent clamped edges exist. The validity and accuracy of the present approach are verified by comparison with published analytical and numerical results. Parametric studies are performed to investigate the influences of nonlocal parameter and boundary conditions on the size-dependency of natural frequencies and critical buckling loads of nanoplates. [ABSTRACT FROM AUTHOR]

This study investigates the size-dependent buckling behavior of Timoshenko nanobeams resting on Winkler-Pasternak foundations in a non-uniform thermal environment. A non-uniform temperature distribution is established through nonlocal heat conduction. Subsequently, the equivalent thermal load due to the obtained temperature distribution and boundary constraints is derived using the governing equations of the axial thermal deformation of the beams based on the stress-driven nonlocal elastic model. To obtain the critical buckling load of the nanobeams, the quadrature element method is used to numerically resolve the eigenvalue problem. In the numerical simulation section, we have presented a series of examples to analyze the effects of length-to-height ratios, nonlocal scale parameters, and Winkler-Pasternak foundation parameters on the buckling loads of the nanobeams under various boundary conditions. Moreover, we examine the effect of comprehensively considering both the elastic and thermal nonlocality on the thermal loads and finally mechanical buckling loads. [ABSTRACT FROM AUTHOR]

We consider mechanically-induced pattern formation within the framework of a growing, planar, elastic rod attached to an elastic foundation. Through a combination of weakly nonlinear analysis and numerical methods, we identify how the shape and type of buckling (super- or subcritical) depend on material parameters, and a complex phase-space of transition from super- to subcritical is uncovered. We then examine the effect of heterogeneity on buckling and post-buckling behaviour, in the context of a heterogeneous substrate adhesion, elastic stiffness, or growth. We show how the same functional form of heterogeneity in different properties is manifest in a vastly differing post-buckled shape. Finally, a fourth form of heterogeneity, an imperfect foundation, is incorporated and shown to have a more dramatic impact on the buckling instability, a difference that can be qualitatively understood via the weakly nonlinear analysis.

Size-dependent functionally graded material thin plate buckling and post-buckling problems are considered using the framework of the MCST (Modified Couple Stress Theory). Based on modified couple stress theory and power law, the post-buckling deflection and critical buckling load of simply supported functionally graded material thin plate are derived using Hamilton's minimum potential energy principle. The analysis compares the simulation results of linear buckling and nonlinear buckling. Innovatively, a power-law distribution with scale effects is considered. The influences of scale effect parameters l and power-law index parameters k on buckling displacement, load, and strain energy of plates have been investigated. In this article, it is found that the critical buckling displacement, critical buckling load, and buckling strain energy increase with increases in the power-law index parameters k. The membrane energy decreases as the power-law index parameter increases. If the upper and lower layers are swapped, the opposite result is obtained. In comparison, the scale effect parameter is more influential than the power-law exponent. The critical buckling displacement in the x-direction is not affected by scale effects. The critical buckling load, the membrane energy, and buckling strain energy increase as the scale effect parameter increases. Scale effects increase material stiffness compared with traditional theory, and the power-law index parameters affect FGM properties such as elastic modulus, Poisson's ratio, density, etc. Both scale effects parameters and power-law index parameters have important effects on the mechanical behavior of materials. [ABSTRACT FROM AUTHOR]

Buckling analysis of functionally graded piezoelectric nanoshell is studied in this paper based on the higher-order shear and normal deformation theory and accounting thickness stretching effect. The nanoshell is subjected to axial load, applied electric potential and thermal loads. Thickness stretching effect is accounted in the analysis based on higher-order shear and normal deformation theory. Small scale effects are accounted based on the Eringen nonlocal elasticity theory. The Navier solution is used for the buckling analysis of the cylindrical nanoshell with simply-supported boundary conditions. The accuracy and trueness of the present paper is justified using comparison with literature. The importance of the present analysis and corresponding results is justified using presentation of results with and without thickness stretching effect. A large parametric analysis is presented to investigate the influence of significant parameters such as dimensionless small scale parameter, length to radius ratio, thickness to radius ratio, temperature rising and applied electric voltage on critical buckling axial loads. One can conclude that the critical buckling axial loads are decreased with increase of small scale parameters and applied electric potential. [ABSTRACT FROM AUTHOR]

Buckling plays a critical role in the transport and dynamics of elastic microfilaments in Stokesian fluids. However, previous work has only considered filaments with homogeneous structural properties. Filament backbone stiffness can be non-uniform in many biological systems like microtubules, where the association and disassociation of proteins can lead to spatial and temporal changes into structure. The consequences of such non-uniformities in the configurational stability and transport of these fibers are yet unknown. Here, we use slender-body theory and Euler-Bernoulli elasticity coupled with various non-uniform bending rigidity profiles to quantify this buckling instability using linear stability analysis and Brownian simulations. In shear flows, we observe more pronounced buckling in areas of reduced rigidity in our simulations. These areas of marked deformations give rise to differences in the particle extra stress, indicating a non-trivial rheological response due to the presence of these filaments. The fundamental mode shapes arising from each rigidity profile are consistent with the predictions from our linear stability analysis. Collectively, these results suggest that non-uniform bending rigidity can drastically alter fluid-structure interactions in physiologically relevant settings, providing a foundation to elucidate the complex interplay between hydrodynamics and the structural properties of biopolymers.

We investigate the effect of defect geometry in dictating the sensitivity of the critical buckling conditions of spherical shells under external pressure loading. Specifically, we perform a comparative study between shells containing dimpled (inward) versus bumpy (outward) Gaussian defects. The former has become the standard shape in many recent shell-buckling studies, whereas the latter has remained mostly unexplored. We employ finite-element simulations, which were validated previously against experiments, to compute the knockdown factors for the two cases while systematically exploring the parameter space of the defect geometry. For the same magnitudes of the amplitude and angular width of the defect, we find that shells containing bumpy defects consistently exhibit significantly higher knockdown factors than shells with the more classic dimpled defects. Furthermore, the relationship of the knockdown as a function of the amplitude and the width of the defect is qualitatively different between the two cases, which also exhibit distinct post-buckling behavior. A speculative interpretation of the results is provided based on the qualitative differences in the mean-curvature profiles of the two cases. [ABSTRACT FROM AUTHOR]

Purpose: In this paper, based on the density functional theory (DFT) and finite element method (FEM), the elastic, buckling and vibrational behaviors of the monolayer bismuthene are studied. Design/methodology/approach: The computed elastic properties based on DFT are used to develop a finite element (FE) model for the monolayer bismuthene in which the Bi-Bi bonds are simulated by beam elements. Furthermore, mass elements are used to model the Bi atoms. The developed FE model is used to compute Young's modulus of monolayer bismuthene. The model is then used to evaluate the buckling force and fundamental natural frequency of the monolayer bismuthene with different geometrical parameters. Findings: Comparing the results of the FEM and DFT, it is shown that the proposed model can predict Young's modulus of the monolayer bismuthene with an acceptable accuracy. It is also shown that the influence of the vertical side length on the fundamental natural frequency of the monolayer bismuthene is not significant. However, vibrational characteristics of the bismuthene are significantly affected by the horizontal side length. Originality/value: DFT and FEM are used to study the elastic, vibrational and buckling properties of the monolayer bismuthene. The developed model can be used to predict Young's modulus of the monolayer bismuthene accurately. Effect of the vertical side length on the fundamental natural frequency is negligible. However, vibrational characteristics are significantly affected by the horizontal side length. [ABSTRACT FROM AUTHOR]

Closed form solutions for the flexural-torsional buckling of elastic beam-columns may only be obtained for simple end boundary conditions, and the case of uniform bending and compression. Moment gradient cases need approximate analytical or numerical methods to be used. Investigations presented in this paper deal with the analytical energy method applied for any asymmetric transverse loading case that produces a moment gradient. Part I of this paper is devoted entirely to the theoretical investigations into the energy based out-of-plane stability formulation and its general solution. For the convenience of calculations, the load and the resulting moment diagram are presented as a superposition of two components, namely the symmetric and antisymmetric ones. The basic form of a non-classical energy equation is developed. It appears to be a function dependent upon the products of the prebuckling displacements (knowfrom the prebuckling analysis) and the postbuckling deformation state components (unknowns enabling the formulation of the stability eigenproblem according to the linear buckling analysis). Firstly, the buckling state solution is sought by presenting the basic form of the non-classical energy equation in several variants being dependent upon the approximation of the major axis stress resultant MY and the buckling minor axis stress resultant MZ. The following are considered: the classical energy equation leading to the linear eigenproblem analysis (LEA), its variant leading to the quadratic eigenproblem analysis (QEA) and the other non-classical energy equation forms leading to nonlinear eigenproblem analyses (NEA). The novel forms are those for which the stability equation becomes dependent only upon the twist rotation and its derivatives. Such a refinement is allowed for by using the second order out-of-plane bending differential equation through which the minor axis curvature shape is directly related to the twist rotation shape. Secondly, the effect of coupling of the in-plane and out-of-plane buckling forms is taken into consideration by introducing approximate second order bending relationships. The accuracy of the classical energy method of solving FTB problems is expected to be improved for both H- and I-section beam-columns. The outcomes of research presented in this part are utilized in Part II. [ABSTRACT FROM AUTHOR]

The present research aims at determining the axial buckling load of stiffened multilayer cylindrical shell panels made of functionally graded graphene-reinforced composites (FG-GPL RCs). Rings and stringers are applied as stiffening tools for shell panels, whose elastic properties are determined according to the Halpin–Tsai relations. The virtual work principle and finite element approach are implemented here, according to a first-order shear deformation theory (FSDT) and Lekhnitskii smeared stiffener approach, in order to determine the governing equations of the stability problem. Four different dispersions of nanofillers are assumed in the thickness direction, including the FG-X, FG-A, FG-O, and UD distributions. A large systematic investigation considers the effect of different geometric and material parameters on the buckling loads and mode shapes of the stiffened FG-GPL RC cylindrical shell panel, primarily the dispersion and weight fractions of the nanofiller, the number of rings and stringers, and the boundary conditions, with useful insights for design purposes. [ABSTRACT FROM AUTHOR]

We identify a new structure-property connection in the skeletal elements of the marine sponge Tethya aurantia. The skeletal elements, known as spicules, are millimeter-long, axisymmetric, silica rods that are tapered along their lengths. Mechanical designs in other structural biomaterials, such as nacre and bone, have been studied primarily for their benefits to toughness properties. The structure-property connection we identify, however, falls in the entirely new category of buckling resistance. We use computational mechanics calculations and information about the spicules' arrangement within the sponge to develop a structural mechanics model for the spicules. We use our structural mechanics model along with measurements of the spicules' shape to estimate the load they can transmit before buckling. Compared to a cylinder with the same length and volume, we predict that the spicules' shape enhances this critical load by up to 30%. We also find that the spicules' shape is close to the shape of the column that is optimized to transmit the largest load before buckling. In man-made structures, many strategies are used to prevent buckling. We find, however, that the spicules use a completely new strategy. We hope our discussion will generate a greater appreciation for nature's ability to produce beneficial designs.

Interest in components with detailed structures increased with the progress in advanced manufacturing techniques. Parts with lattice elements can provide improved global buckling stability compared to solid structures of the same weight. However, thin features are prone to local buckling. We present a two-scale optimization approach that simultaneously improves the local and global stability of parametrized graded lattice structures. Elastic properties and local buckling behavior are upscaled via homogenization based on geometric exact beam theory. To reduce computational effort, we construct a worst-case model for the homogenized buckling load factor, which acts as a safeguard against local buckling. We briefly discuss advantages and limitations by means of numerical examples. [ABSTRACT FROM AUTHOR]

Curved nanobeams are one of the essential components in manufacturing nano-electromechanical systems needing nonlinear stability design. In the current investigation, the nonlinear buckling characteristics of functionally graded porous reinforced curved (FGPRC) nanobeams having different degrees of curvature are analyzed by counting the higher-order gradients of the classical strain tensor as well as nonlocal-type interatomic interactions. In this regard, two independent length-scale constants within the framework of the nonlocal strain gradient theory (NSGT) of continuum elasticity are taken into account. Via employing the promising low computational cost and geometrically adaptable method of isogeometric collocation, various branches of NSGT-based equilibrium graphs of FGPRC nanobeams are plotted relevant to each considered degree of curvature. It is extrapolated that the quantity of graphene platelet (GPL) weight fraction has a negligible influence on the significance of nonlocal-type of interatomic size dependency as well as the strain gradient kind of small-scale effect in the value of the maximum deflections or lateral loads at the detected limit points, especially attributed to a higher degree of curvature. Besides, it can be remarked that, in the FGPRC nanobeam owning a small degree of curvature, lessening the quantity of porosity index results in an increment in the significance of the nonlocal-type of interatomic size dependency as well as the strain gradient kind of small-scale effect on the maximum deflection at both upper and lower limit points. However, in the FGPRC nanobeams owning medium and large degrees of curvature, it gets lesser at the upper limit point, but becomes higher at the lower limit one. [ABSTRACT FROM AUTHOR]

Although it is often asserted that, in view of their reduced length, axially compressible beams have a higher buckling load than their inextensible counterparts, a detailed analysis demonstrates that this is not necessarily the case. The argument to arrive at this conclusion is made in terms of relatively straightforward concepts of elasticity and structural mechanics. It is shown that for certain classes of materials, the reduced prebuckling length is more than compensated for by a softening of the elastic response, leading to a reduction of the Euler critical load. [ABSTRACT FROM AUTHOR]

The numerical response of one-storey reinforced concrete frames designed following design standards before the enactment of modern seismic provisions was investigated in this study using the nonlinear finite element method. An unbraced frame and two frames retrofitted with a buckling restrained brace (BRB) incorporating either a stainless steel or chrome molybdenum core bar were modelled. The nonlinear static reverse cyclic loading performance of the frames was assessed. Modifications to the BRB are proposed to mitigate deficiencies identified with the original BRB and steel core bars. The modified BRB incorporating a superelastic shape-memory alloy (SE-SMA) core bar was further modelled. The results illustrate the benefits of implementing SE-SMA as a retrofit strategy to control permanent displacements. The modified BRB with a SE-SMA core exhibited improved lateral strength and displacement capacities relative to the frames retrofitted with steel cores and the control bare frame. [ABSTRACT FROM AUTHOR]

Buckling of a simple supported and cantilever nano‐rod and under action of self‐weight was modeled, taking into account the surface effect according to two models of surface elasticity – Gurtin–Murdoch and Steigmann–Ogden. The size effect under the buckling was studied by the Ritz method using two kinematic hypotheses, known as the rod theories of Euler–Bernoulli and Timoshenko. The simulation results were compared depending on the accepted surface elasticity model and the kinematic hypothesis. [ABSTRACT FROM AUTHOR]

• A new exact buckling model based on MGE for thin-walled microbeams is proposed. • Besides the classical flexural–torsional coupling, two new coupling effects are found. • The first new coupling effect is the higher-order flexure–flexure coupling. • The second new coupling effect is the higher-order flexural–torsional coupling. More and more thin-walled micro-beams, including the I-section thin-walled micro-beams, are applied in MEMS. The buckling of a thin-walled micro-beam exhibits some different phenomena from that of a large scale thin-walled beam, such as the size effect, and the coupling effects caused by strain gradient. To investigate the influence of the coupling effects and size effect on the buckling of thin-walled micro-beams, a new buckling model is proposed. Based on modified gradient elasticity (MGE), the governing equations and boundary conditions of the new model are derived from the variational principle. The governing equations and boundary conditions can be simplified to the classical Vlasov's theory, the MGE Bernoulli–Euler beam model, MGE Bernoulli–Euler beam model considering bi-directional flexure respectively. Solved by the feature expansion method, the size effects of the thin-walled micro-beam can be captured by the new model, that is the buckling load increases with internal length scales. Besides the classical flexural–torsional coupling effect, two new coupling effects, such as the higher-order flexure–flexure coupling and the higher-order flexural–torsional coupling, are also captured by the new model. These two new high-order coupling effects of the new model reduce the critical buckling load of thin-walled micro-beams. The higher-order flexural–flexural coupling effect is far greater than the other two coupling effects, and the higher-order flexural–flexural coupling effect is on the same order of magnitude as the effect of MGE (size effect), with the former being negative and the latter positive. The size effect and higher-order coupling effect cannot be ignored and need to be considered carefully for the buckling of thin-walled micro-beams. [Display omitted] [ABSTRACT FROM AUTHOR]

This paper presents a first investigation of the mechanical behavior of porous functionally-graded material (FGM) plates reinforced with graphene platelets (GPls) nanomaterial. As a main purpose, we extend the research on GPls incorporation from single-component matrix materials to porous metal-ceramic FGMs. Four different through-thickness distributions of GPls and porosities were investigated. The effective elastic modulus of GPls-reinforced FGM was calculated using the Halpin-Tsai homogenization model, and the effect of porosities on elastic properties was considered by an exponential model that is more accurate than the common rule of mixture. A recent hyperbolic higher-order plate theory that satisfies the traction-free boundary conditions and a parabolic variation of shear stresses was used to model kinematics. The influence of various parameters was studied after method validation, and it was revealed that GPls not only can improve the stiffness and mechanical response of FGM plates, but can also eliminate the inherent transverse anisotropy of FGMs and uncouple the static response when dispersed in proper distribution pattern. This effect will result in zero mid-plane extension in bending conditions, in a similar behavior to isotropic plates. [ABSTRACT FROM AUTHOR]

We have investigated the mechanism of formation of the body of a typical vertebrate, the chicken. We find that the body forms initially by folding at boundaries of stiffness contrast. These boundaries are dynamic lines, separating domains of different cell sizes, that are advected in a deterministic thin-film visco-elastic flow. While initially roughly circular, the lines of elastic contrast form large "peanut" shapes evoking a slender figure-8 at the moment of formation of the animal body, due to deformation and flow in a quadrupolar stretch caused by mesoderm migration. Folding of these "peanut" or "figure-8" motives along the lines of stiffness contrast creates the global pattern of the animal, and segregates several important territories. The main result is that the pattern of cell texture in the embryo serves simultaneously two seemingly different purposes: it regionalizes territories that will differentiate to different cell types and it also locks the folds that physically segregate these territories. This explains how the different cellular types segregate in physically separated domains.

To accurately capture size effect and geometrically nonlinear effect on thermal post-buckling of micro-beams, a new theoretical analysis model is presented. The model is based on modified gradient elasticity and von Kármán geometrically nonlinear theory. Considering the thermal effect, the governing equations of thermal post-buckling are derived by the principle of minimum total potential energy. The complete information including post-buckling bifurcation, configuration, amplitude of hinged–hinged and clamped–clamped micro-beams under different temperature rises is obtained by analytical solution. Comprehensive discussions are represented for size effect and geometrically nonlinear effect on thermal post-buckling of micro-beams. It confirms that taking into account the effect of geometric nonlinearity results in smaller post-buckling amplitude and the buckling resistance increase with the decrease in beam size. Both size effect of the critical buckling temperature rises and the post-buckling amplitude can be captured. Compared with other models, the ability of this model to predict the thermal buckling behaviors is improved. [ABSTRACT FROM AUTHOR]

Motivated by the observations of snap-through phenomena in pre-stressed strips and curved shells, we numerically investigate the snapping of a pre-buckled hemispherical gridshell under apex load indentation. Our experimentally validated numerical framework on elastic gridshell simulation combines two components: (i) discrete elastic rods method, for the geometrically nonlinear description of one-dimensional rods, and (ii) a naive penalty-based energy functional, to perform the non-deviation condition between two rods at joint. An initially planar grid of slender rods can be actuated into a three-dimensional hemispherical shape by loading its extremities through a prescribed path, known as buckling-induced assembly; next, this pre-buckled structure can suddenly change its bending direction at some threshold points when compressing its apex to the other side. We find that the hemispherical gridshell can undergo snap-through buckling through two different paths based on two different apex loading conditions. The structural rigidity increases as the number of rods in the gridshell structure becomes denser, which emphasizes the mechanically nonlocal property in hollow grids, in contrast to the local response of continuum shells. The findings may bridge the gap among rods, grids, knits, and shells, for a fundamental understanding of a group of thin elastic structures, and inspire the design of novel micro-electro-mechanical systems and functional metamaterials. [ABSTRACT FROM AUTHOR]

The size-dependent buckling problem of cracked micro- and nanocantilevers, which have many applications as sensors and actuators, is studied by the stress-driven nonlocal theory of elasticity and Bernoulli–Euler beam model. The presence of the crack is modeled by assuming that the sections at the left and right sides of the crack are connected by a rotational spring. The compliance of the spring, which relates the slope discontinuity and the bending moment at the cracked cross section, is related to the crack length using the method of energy consideration and the theory of fracture mechanics. The buckling equations of the left and right sections are solved separately, and the variationally consistent and constitutive boundary and continuity conditions are imposed to close the problem. Novel insightful results are presented about the effects of the crack length and location, and the nonlocality on the critical loads and mode shapes, also for higher modes of buckling. The results of the present model converge to those of the intact nanocantilevers when the crack length goes to zero and to those of the large-scale cracked cantilever beams when the nonlocal parameter vanishes. [ABSTRACT FROM AUTHOR]

The need to improve on analysis and design of structural members for overall weight, material and financial economy is indispensable. This paper presents the results of post-buckling analysis of an axially loaded prismatic column. The theory of elasticity was employed to formulate the total potential energy functional for a prismatic column in the post buckling regime. The derived functional was minimized with respect to deflection function to obtain the governing equation of equilibrium. The governing equation was then solved to obtain the deflection equation. Again, the functional was minimized with respect to the coefficient of the obtained deflection function to obtain the formula for calculating the post-buckling load of the prismatic column. A numerical example was given to demonstrate the applicability of the present formulation. The prismatic column was analyzed within the post buckling regime to determine the stiffness coefficients. It was shown that as the deflection-thickness (Δ/t) ratio increased, the combined buckling stress increased geometrically. It was found that the critical buckling load obtained from the present method and those obtained from literature showed consistency. It was also found that at the buckled state of the prismatic column, the yield strength of the column material has not been attained and deflection is very small. This gives room for the column to take on more loads before it fails either by excessive deflection or by stress exceeding the material strength. [ABSTRACT FROM AUTHOR]

The results of an experimental study on the buckling of a vertically cantilevered plate under corner twisting forces are reported. In this configuration, an interesting and somewhat counter-intuitive behavior is observed in which a laterally loaded slender panel exhibits a subtle instability characterized by nonlinear out-of-plane corner deflections. This is one of the few studies in the literature that investigates the buckling of cantilevered plates and appears to be the only experimental study of buckling under twisting loads, i.e., loading the plate with point loads at the free corners and in opposite directions. This paper discusses the practical aspects of experimental verification and methodology and evaluates the effect of the plate aspect ratio (AR) on the nondimensional buckling load in this configuration. In addition, experimental results are compared to finite element analysis (FEA) simulations performed on accessible software. There is generally good agreement between the experiment and the finite element simulation for the change of buckling load with plate aspect ratio. The nondimensional buckling load appears to decrease asymptotically with increasing aspect ratio, though the correlation between experiment and computation is less consistent as aspect ratio increases due to experimental and computational limitations. [ABSTRACT FROM AUTHOR]

When an overhand knot tied in an elastic rod is tightened, it can undergo a sudden change in shape through snap buckling. In this article, we use a combination of discrete differential geometry (DDG)-based simulations and tabletop experiments to explore the onset of buckling as a function of knot topology, rod geometry, and friction. In our setup, two open ends of an overhand knot are slowly pulled apart, which leads to snap buckling in the knot loop. We call this phenomenon "inversion" since the loop appears to move dramatically from one side of the knot to the other. This inversion occurs due to the coupling of elastic energy between the braid (the portion of the knot in self-contact) and the loop (the portion of the knot with two ends connected to the braid). A numerical framework is implemented that combines discrete elastic rods with a constraint-based method for frictional contact to explore inversion in overhand knots. The numerical simulation robustly captures inversion in the knot and is found to be in good agreement with experimental results. In order to gain physical insight into the inversion process, we also develop a simplified model of the knot that does not require simulation of self-contact, which allows us to visualize the bifurcation that results in snap buckling. [ABSTRACT FROM AUTHOR]

The buckling response of functionally graded (FG) porous spherical caps reinforced by graphene platelets (GPLs) is assessed here, including both symmetric and uniform porosity patterns in the metal matrix, together with five different GPL distributions. The Halpin–Tsai model is here applied, together with an extended rule of mixture to determine the elastic properties and mass density of the selected shells, respectively. The equilibrium equations of the pre-buckling state are here determined according to a linear three-dimensional (3D) elasticity basics and principle of virtual work, whose solution is determined from classical finite elements. The buckling load is, thus, obtained based on the nonlinear Green strain field and generalized geometric stiffness concept. A large parametric investigation studies the sensitivity of the natural frequencies of FG porous spherical caps reinforced by GPLs to different parameters, namely, the porosity coefficients and distributions, together with different polar angles and stiffness coefficients of the elastic foundation, but also different GPL patterns and weight fractions of graphene nanofillers. Results denote that the maximum and minimum buckling loads are reached for GPL-X and GPL-O distributions, respectively. Additionally, the difference between the maximum and minimum critical buckling loads for different porosity distributions is approximately equal to 90%, which belong to symmetric distributions. It is also found that a high weight fraction of GPLs and a high porosity coefficient yield the highest and lowest effects of the structure on the buckling loads of the structure for an amount of 100% and 12.5%, respectively. [ABSTRACT FROM AUTHOR]

[Display omitted] • Buckling model of an active plate is established based on the odd elasticity theory. • Anomalous tensile buckling of odd elastic plates is investigated. • Novel form of active buckling has been identified. • Active buckling is concluded to only occur in the form of chiral deformation. Buckling of thin-walled structures such as plates and shells is a consequence of in-plane stress being released through out-of-plane displacements. Generally, thin-walled structures that are subjected to tension or zero stress conditions remain stable. In this article, the anomalous tension buckling and stress-free active buckling of odd elastic plates are reported, which are a novel instability caused by odd elastic effects. The latter can only occur in the form of left- or right-handed chiral deformation and it does not involve external loads or internal active stress in a critical state. We demonstrate that the chiral rotation angle deformation is responsible for the active buckling of the plates, because the energy required for instability can be obtained based on the odd elastic effect. These findings can serve as an interpretation in a novel way for the occurrence of surface morphologies with biological activities, as well as provide references for buckling designs and applications of active structures. [ABSTRACT FROM AUTHOR]

A mathematical model for nonlocal vibration and buckling of embedded two-dimensional (2D) decagonal quasicrystal (QC) layered nanoplates is proposed. The Pasternak-type foundation is used to simulate the interaction between the nanoplates and the elastic medium. The exact solutions of the nonlocal vibration frequency and buckling critical load of the 2D decagonal QC layered nanoplates are obtained by solving the eigensystem and using the propagator matrix method. The present three-dimensional (3D) exact solution can predict correctly the nature frequencies and critical loads of the nanoplates as compared with previous thin-plate and medium-thick-plate theories. Numerical examples are provided to display the effects of the quasiperiodic direction, length-to-width ratio, thickness of the nanoplates, nonlocal parameter, stacking sequence, and medium elasticity on the vibration frequency and critical buckling load of the 2D decagonal QC nanoplates. The results show that the effects of the quasiperiodic direction on the vibration frequency and critical buckling load depend on the length-to-width ratio of the nanoplates. The thickness of the nanoplate and the elasticity of the surrounding medium can be adjusted for optimal frequency and critical buckling load of the nanoplate. This feature is useful since the frequency and critical buckling load of the 2D decagonal QCs as coating materials of plate structures can now be tuned as one desire. [ABSTRACT FROM AUTHOR]

This paper proposes a modified segmented rod method (MSRM) in order to facilitate the introduction of boundary conditions in the analysis of free vibrations and buckling of Euler–Bernoulli beams. First, the basic idea of proposed modifications is presented in the case of elementary length beam, and then the idea is generalized to the full-length beam. In addition to facilitating the introduction of boundary conditions, by comparing the results of MSRM with segmented rod method (SRM) and Hencky bar-chain method (HBM), a faster convergence of MSRM results to the exact ones was observed. Although the case of uniform beam is considered in this paper, proposed MSRM may contribute to greater use of segmented beam approach in solving of various static, free vibration or buckling problems of non-uniform beams. Also, it was shown that MSRM can be easily adapted for the analysis of beams that resting on partial elastic foundation. [ABSTRACT FROM AUTHOR]

The need to improve on analysis and design of structural members for overall weight, material and financial economy is indispensable. This paper presents the results of post-buckling analysis of an axially loaded prismatic column. The theory of elasticity was employed to formulate the total potential energy functional for a prismatic column in the post buckling regime. The derived functional was minimized with respect to deflection function to obtain the governing equation of equilibrium. The governing equation was then solved to obtain the deflection equation. Again, the functional was minimized with respect to the coefficient of the obtained deflection function to obtain the formula for calculating the post-buckling load of the prismatic column. A numerical example was given to demonstrate the applicability of the present formulation. The prismatic column was analyzed within the post buckling regime to determine the stiffness coefficients. It was shown that as the deflection-thickness (Δ/t) ratio increased, the combined buckling stress increased geometrically. It was found that the critical buckling load obtained from the present method and those obtained from literature showed consistency. It was also found that at the buckled state of the prismatic column, the yield strength of the column material has not been attained and deflection is very small. This gives room for the column to take on more loads before it fails either by excessive deflection or by stress exceeding the material strength. [ABSTRACT FROM AUTHOR]

Long glass fiber-reinforced polypropylene (LGFR-PP) composite structures with stiffeners are important substitutes for metal parts for vehicle lightweighting; a good understanding of the buckling characteristics of LGFR-PP stiffeners would provide an important reference for engineering design. The current work is therefore intended to study the buckling characteristics of different cross-sectioned LGFR-PP stiffeners under axial compression via experimental and theoretical analysis. Firstly, LGFR-PP stiffeners with semicircular, rectangular, and trapeziform cross-sections were compressed at the axial direction using a universal testing machine to obtain the buckling process data. Then, the elasticity stability theory modified according to the experimental results was derived to estimate the buckling resistance of LGFR-PP stiffeners in different designs. The test results showed that the LGFR-PP stiffeners possessed a flexible–torsional bulking instability mode under axial compression, the LGFR-PP stiffeners with a semicircular cross-section had higher compression buckling resistance, and the rectangular and trapeziform cross-sectioned stiffeners had better rigidity. The theoretical analysis showed that the modified elasticity stability theory could generally predict the buckling resistance of LGFR-PP stiffeners under axial compression. [ABSTRACT FROM AUTHOR]

In this paper, we present an analytical method based on equivalent continuum homogenization for the global buckling analysis of the general grid lattice cylindrical structures. In the proposed scheme, grid structures with arbitrary cell geometries can be analyzed by obtaining their effective cell stiffness based on force and moment analysis of the struts. The grid structures are assumed to be composed of curvilinear axial, circumferential, and oblique unidirectional composite ribs. To evaluate the results of the presented analytical method, a parametric finite element code is derived to generate the desired geometry of the grid structures and their buckling loads are obtained and compared with the analytical method. The effects of various parameters, including the number of ribs, their thickness and elastic properties, and helical angle of the oblique ribs are studied for hexagonal, triangular, and mixed grid shells. The results are compared with the available data published in the literature with a good agreement. [ABSTRACT FROM AUTHOR]

The thermal buckling load factors of damaged (crack) laminated composite structures have been investigated numerically first-hand. Their strength reversal due to the functional material reinforcement under the elevated environmental conditions is reported in detail. In this regard, a generic finite element model of the damaged layered structure has been derived in the higher-order kinematic model considering the Green–Lagrange type of strain (to count the large deformation due to temperature). Further, the shape memory alloy (SMA) fibre effect has been introduced in the proposed mathematical model via marching technique under the change in temperature to achieve the modified elastic property. The final governing equation of buckled structure has been derived and solved through isoparametric finite element steps. The final buckling temperatures of the damaged layered structural system with and without SMA (volume fraction and prestrain) fibres were obtained, and the results indicate a good improvement (maximum up to 37%). Additionally, the structural geometry-related parameters, including their shapes, have been changed to show the model's applicability. [ABSTRACT FROM AUTHOR]

Highlights • A finite element is developed for lateral torsional buckling analysis of wood beams. • Captures partial twist restraint, load height, and pre-buckling deformation effects. • Solution is validated against 3D finite element modelling and experimental results. • Simplified analytical expressions are developed to estimate critical moments. Abstract A beam finite element formulation is developed for the elastic lateral torsional buckling analysis of wood beams with rectangular cross-sections. The formulation accounts for moment gradient, load height, and pre-buckling deformation effects. The validity of the finite element and underlying variational principle is established through comparisons with critical moments obtained by 3D FEA simulations and experimental results. The validated variational expression is then adopted to develop approximate analytical expressions that separately characterize the effects of moment gradient, partial twist restraints that may exist at beam-ends, load height, and pre-buckling deformation effects. Dimensionless coefficients were formulated to characterize each effect for simple beams and cantilevers under common loading scenarios. The derived coefficients are then consolidated into a unified framework that accounts for all effects combined to predict the elastic lateral torsional buckling capacity of wooden beams. [ABSTRACT FROM AUTHOR]

In the present work, the small size effects on stability properties of perforated microbeams under various types of deformable boundary conditions are studied considering the Fourier sine series solution procedure and a mathematical procedure known as Stokes' transformation for the first time. The main benefit of the present method is that, in addition to considering both the gradient elasticity and the size effects, the kinematic boundary conditions are modeled by two elastic springs as deformable boundary conditions. The deformable boundary conditions and corresponding stability equation are described using the classical principle which are then used to construct a linear system of equations. Afterward, an eigenvalue problem is adopted to obtain critical buckling loads. The correctness and accuracy of the present model are demonstrated by comparing results with those available from other works in the literature. Moreover, a numerical problem is solved and presented in detail to show the influences of the perforation properties, geometrical, and the variation of small-scale parameters and foundation parameters on the stability behavior of the microbeams. In addition, according to the best knowledge of the authors, there is no study in the literature that examines the buckling behavior of perforated microbeams on elastic foundation with the gradient elasticity theory. [ABSTRACT FROM AUTHOR]

Abstract The stability of a spinning thin elastic disk has been widely studied due to its central importance in engineering. While the plastic deformation and failure of an annular disk mounted on a rigid and accelerating circular shaft are well understood, shear-induced elastic buckling of the disk due to this 'spin-up' is yet to be reported. Here, we calculate this buckling behavior within the framework of the Föppl–von Kármán equations and give numerical results as a function of the disk's aspect ratio (inner-to-outer radius) and Poisson's ratio. This shows that shear-induced elastic buckling can dominate plastic failure in many cases of practical interest. When combined with existing theory for plastic failure, the results of the present study provide foundation results for a multitude of applications including the characterization of accelerating compact disks and deployment of space sails by centrifugal forces. [ABSTRACT FROM AUTHOR]

Buckling analysis of a functionally graded (FG) nanobeam resting on two-parameter elastic foundation is presented based on third-order shear deformation beam theory (TOSDBT). The in-plane displacement of TOSDBT has parabolic variation through the beam thickness. Also, TOSDBT accounts for shear deformation effect and verifies stress-free boundary conditions on upper and lower faces of FG nanobeam. The twoparameter elastic foundation model including linear Winkler's springs along with Pasternak's shear layer is in contact with the beam in deformation. Material properties of FG nanobeam are supposed to vary gradually along the thickness according to both power-law and Mori-Tanaka laws. Small-scale effect of Eringen's nonlocal elasticity theory has been considered. Nonlocal equilibrium equations are obtained based on the minimum potential energy principle and solved analytically. The accuracy of current method is examined by comparing current results with the available ones in literature. Effects of foundation parameters, gradient index, nonlocal parameter and slenderness ratio on buckling behavior of FG nanobeams are examined. [ABSTRACT FROM AUTHOR]

Abstract Mechanical behavior of oilwell drillstring coming at the motions of penetrating rock is associated with drilling safety and wellbore quality. A poor performance can be attended by elastic stability loss in a slender drillstring exerting undesired buckling and vibration phenomena in the working process. Devoting to investigating the dynamic buckling behavior is an efficient way to avoid fatigue failure of the drillstring components and enhance drilling rate. A dynamic analysis of the buckling behavior of the entire column considering violent lateral vibrations is presented as a result available. Analytical equations of the drilling system are integrated numerically with finite element method. A virtual contact element implemented between the drillstring and the borehole, in conjunction with Hertzian contact assumption, gives the direct access to calculate the contact force and positions. Lateral deflection, critical length and the effects of drilling parameters and stabilizer on the buckling deformation are captured to experience the realistic buckling response of a typical drillstring in a vertical well. Simulation results indicate that spiral pitch and wave numbers of a nonlinear spiral shape prevailing in the lower zone of drillstring are deeply associated with the cutting force at the bit. Improving a drillstring behavior performance in a real oilfield is observed that this model can assist in evaluating the properties of the working security and buckling resistance with complex boundary conditions. Highlights • Mechanical behavior affects oilwell drilling safety and wellbore quality. • Predict buckling behavior can avoid drillstring fatigue failure and enhance drilling rate. • A dynamic approach is proposed to study the buckling behavior of the entire drillstring. • Lateral deflection, critical length and contact forces are used to characterize buckling response. • Effects of drilling parameters and stabilizer on buckling deformation provide more behavior details. [ABSTRACT FROM AUTHOR]

Highlights • Low-cycle fatigue life of Grade 300E and 500E reinforcing bars. • Effect of buckling on low-cycle fatigue behaviour of reinforcing bars. • Low-cycle fatigue model incorporating the effect of buckling. • Effect of mean strain ratio on fatigue response of bars. Abstract Low-cycle fatigue life of Grade 300E and 500E reinforcing bars including the effect of buckling is evaluated. Low-cycle fatigue tests are carried out on un-machined specimens subjected to constant axial strain loading with strain amplitude ranging from 1% to 5%. The experimental test observations suggest that buckling of reinforcing bars has detrimental effect on its fatigue life and increase in slenderness ratio of the reinforcing bars results in substantial reduction of their low-cycle fatigue life. Equations relating the fatigue life (i.e. number of reversals to failure) with total strain amplitude and total energy dissipated as a function of buckling parameter is proposed. Also, experimental tests are carried out to investigate the influence of mean strain ratio on low-cycle fatigue behaviour of reinforcing bars. Comparative evaluation of the fatigue life models with the results of experimental investigation shows that the proposed model can predict the fatigue life of buckled and unbuckled reinforcing bars with reasonable accuracy. [ABSTRACT FROM AUTHOR]

In this article, an analytical approach is presented to study the surface and flexoelectric effects on the buckling characteristics of an embedded piezoelectric sandwich nanobeam. According to the nonlocal elasticity theory, the flexoelectricity is believed to be authentic for size-dependent properties in nanostructures. The boundary conditions and the governing equations are derived by Hamilton's principle and are solved by Navier method. The results obtained from the present work show that the nonlocal term has an important reduction on the critical load and also the flexoelectricity shows an increasing influence on the buckling loads of the sandwich nanobeam, especially at lower thicknesses. [ABSTRACT FROM AUTHOR]

Highlights • A practical, educational augmentation for teaching elastic buckling. • Innovative use of 3D printing. • Measurement of critical loads. • Justification for various theoretical approximations. • Parameter variations and how geometry influences buckling. Abstract This is the third part in a trilogy of papers examining ways in which additive manufacturing can be used to facilitate the introduction of basic principles in structural analysis. Each paper uses 3D-printing and simple, but non-trivial, slender geometric forms, to provide a hands-on aspect to structural behavior in which flexure plays a dominant role. The first part dealt with linear structural analysis (Virgin, 2017), extended to dynamics and vibration in the second part (Virgin, 2017). The current paper focuses on slender structures in which compressive axial loading is the new ingredient and hence buckling becomes a central issue. This has similarities and differences with the two previous papers, but in all instances the role played by relatively high-precision 3D-printing opens the door to versatile and effective illustration, and the development of a deeper appreciation of structural phenomena. [ABSTRACT FROM AUTHOR]

Abstract This study considers the buckling of double nanofibers embedded in an elastic matrix based on an Euler-Bernoulli beam model. A scale dependent doublet mechanics theory is used in modelling of the double nanobeam system. Critical buckling loads are obtained using doublet mechanics and results are compared with the classical elasticity theory. The variation of critical buckling loads with different beam length, doublet separation distance and stiffness of the springs are investigated. Some mode shapes of the double nanobeam system are presented. [ABSTRACT FROM AUTHOR]

This study presents a newly developed size-dependent beam-substrate medium model for bending, buckling, and free-vibration analyses of nanobeams resting on elastic substrate media. The Euler-Bernoulli beam theory describes the beam-section kinematics and the Winkler-foundation model represents interaction between the beam and its underlying substrate medium. The reformulated strain-gradient elasticity theory possessing three non-classical material constants is employed to address the beam-bulk material small-scale effect. The first and second constants is associated with the strain-gradient and couple-stress effects, respectively while the third constant is related to the velocity-gradient effect. The Gurtin-Murdoch surface elasticity theory is adopted to account for the surface-free energy. To obtain the system governing equation as well as corresponding boundary conditions, Hamilton’s principle is called for. Three numerical simulations are presented to characterize the influences of the material small-scale effect, the surface-energy effect, and the surrounding substrate medium on bending, buckling, and free vibration responses of nanobeam-substrate medium systems. The first simulation focuses on the bending response and shows the ability of the proposed model to eliminate the paradoxical characteristic inherent to nanobeam models proposed in the literature. The second and third simulations perform the sensitivity investigation of the system parameters on the buckling load and the natural frequency, respectively. All analytical results reveal that both material small-scale and surface-energy effects consistently stiffen the system response while the velocity-gradient effect weakens the system response. Furthermore, these sized-scale effects are more pronounced when the underlying substrate medium becomes softer. [ABSTRACT FROM AUTHOR]

The classical reduced model for film/substrate bilayers is the one in which the film is governed by the Euler–Bernoulli beam equation, and the substrate is replaced by an array of springs (the so-called Winkler foundation assumption). We derive a refined model in which the normal and shear tractions at the bottom of the film are expressed in terms of the corresponding horizontal and vertical displacements, and the response of the half-space is described by the exact theory. The self-consistency of the refined model is confirmed by showing that it yields a four-term (in the incompressible case) or six-term (in the compressible case) expansion for the critical strain that agrees with the expansion given by the exact theory. [ABSTRACT FROM AUTHOR]