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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]

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 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.

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 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 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]

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]

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]

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]

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]

The elastic buckling of shell structures such as spherical shells subject to external pressure and cylindrical shells loaded in axial compression is highly sensitive to imperfections and often catastrophic. Recent studies of spherical shells have provided accurate quantitative results for the relation between the buckling pressure and the amplitude and shape of geometric imperfections and, additionally, quantitative results for the energy barrier that must be overcome to buckle the shell by extraneous loadings or disturbances when it is loaded to pressures below the buckling pressure. Results for the simultaneous interaction of imperfections and energy barriers for spherical shells under external pressure will be presented. Numerical studies for probing forces illustrate their use in determining the buckling energy barrier, and new experimental results on energy barriers obtained by others by probing spherical shells will be discussed and compared with predictions. It will be argued that while imperfections determine the buckling load of a shell, the energy barrier at loads below the buckling load supplies important additional information about the relative safety or precariousness of the shell to additional disturbances. Results for the energy barrier for perfect and imperfect spherical shells under external pressure provide important insights into the shell's robustness, or lack thereof, at pressures below the buckling pressure. In particular, the energy barrier trends provide critical insights into the low knockdown factor usually employed in establishing the design load of unstiffened spherical and cylindrical shells. These design loads are shown to correlate with conventional predictions provided that imperfection amplitudes scale as the shell radius. [ABSTRACT FROM AUTHOR]

This paper deals with the representative challenging buckling problem of a fully free plate under biaxial compression by a distinctive symplectic superposition method, which yields the benchmark analytic solutions by converting the problem to be solved into the superposition of two elaborated subproblems that are solved by the symplectic elasticity approach. The solution is advanced in the symplectic space-based Hamiltonian system rather than in the classic Euclidean space-based Lagrangian system, which shapes the main advantage of the method that a direct rigorous derivation is qualified for obtaining the analytic solutions, without any assumptions or predetermination of the solution forms. Comprehensive new analytic results for both the buckling loads and mode shapes are presented and validated by the finite element method. The fast convergence and accuracy of the method make it applicable to analytic modeling of more plate problems. [ABSTRACT FROM AUTHOR]

Free vibration and buckling of a set of parallel Timoshenko beams joined by intermediate flexible connections under axial loading are assessed in this paper. The numbers of beams and intermediate connections are arbitrary. Through axial loading, a new set of equations is extracted, applicable in solving the axial loading problems. The existence of the intermediate connections and the related compatibility equations allow for the introduction of coupled partial differential equations. The solution involves a change of variables to uncouple the governing differential equations. The natural frequencies and mode shapes are calculated through the transfer matrix method. By presenting an appropriate equation, it is observed that the general formulation studied here, regarding the separated flexible connections, can easily cover the issue of parallel beams joined by Winkler elastic layers in a continuous manner presented in the literature. The effects of different parameters such as the number of beams, number and stiffness of elastic connections and axial loading are assessed on the natural frequencies and the critical buckling force. [ABSTRACT FROM AUTHOR]

Main concern of the paper is to illustrate the generic application of the optimal control method in the analysis of a constrained buckling problem and its simple and elegant formulation when compared to other techniques. Thus, an optimal control methodology is adopted to investigate the buckling behavior of thin elastic structures under the presence of unilateral constraints. We particularly explore the post-buckling response of a constant or variable length elastica constrained by rigid walls. The equivalence between the calculus of variations and the optimal control is first demonstrated, with a main focus on the post-buckling behavior of a constant length elastica which is constrained by horizontal walls. The gradual construction of the constrained buckling problem as an optimal control problem from its equivalent Lagrangian form clearly shows its advantage when compared to the calculus of variations. The necessary optimality conditions, which constitute the Pontryagin’s minimum or maximum principle, are also derived by considering a direct adjoining approach. The solution of the optimal control problem is then performed by applying a direct method. Validation of the methodology is first achieved by reproducing examples available in the literature. Then the effects of factors, such as the geometry of the walls and the variability in the bending stiffness of the elastica, on its buckling response are analyzed. These constrained buckling problems are investigated for the first time, while through them it is also readily shown that the presence of geometric or material nonlinearity does not introduce any essential complexity in the buckling analysis when the optimal control methodology is adopted. [ABSTRACT FROM AUTHOR]

This work presents an analytical study of the electromechanical buckling of a micro spherical thin film bonded to a compliant elastic substrate. The spherical film is subjected to electrostatic attraction forces that are induced by applying a voltage difference between an outer spherical elastic film electrode and inner rigid, fixed and grounded spherical electrode. When the applied voltage is small, the film contracts while maintaining its complete spherical shape. However, when the applied voltage reaches a critical value, the film buckles into high-ordered periodic patterns (i.e. elastic surface wrinklings). Motivated by experimental results of other studies, this work examines the critical buckling state of one-dimensional, square checkerboard and hexagonal patterns. As will be shown, the above considered patterns are associated with the same critical state, and therefore, all patterns have equal buckling voltage and critical wavelength. Furthermore, with increasing radius of the film/substrate system the electromechanical buckling response converges to the electromechanical pull-in instability of the well-known two parallel plate electrodes, as will be revealed by the asymptotic analytical solution. Finally, the elastic ripples of the electromechanically buckled film can be generated or removed by a simple On/Off switching of the applied voltage. The ability to generate and remove elastic ripples tremendously increases the potential of such microsystem to be utilized in different applications in the field of Micro and Nano electromechanical systems. [ABSTRACT FROM AUTHOR]

Abstract An ultrathin film is flexible but tends to buckle when subjected to compression and temperature variation. The buckling behavior will adversely affect its mechanical performance, therefore, it should be accurately evaluated and under controlled. Accordingly, it is vital to study thermal buckling behavior of ultrathin films. In the present work, thermal buckling of bilayer graphene sheets (GSs) embedded in Pasternak-type foundations is studied based on the nonlocal elastic theory and classical plate theory (CLPT). We have examined three types of thermal distribution, namely linear, nonlinear and uniform temperature distributions through the thickness of GSs. The effects of boundary condition, aspect ratio, nonlocal parameter, elastic foundation parameter, geometric size, stacking types on the critical buckling temperature loads are investigated. [ABSTRACT FROM AUTHOR]

In this paper, an analytical Hamiltonian-based model for the dynamic analysis of rectangular nanoplates is proposed using the Kirchhoff plate theory and Eringen's nonlocal theory. In a symplectic space, the dynamic problem is reduced to solving a unified Hamiltonian dual equation formed by a total unknown vector consisting of displacements, rotation angles, bending moments and generalized shear forces. The exact solutions for free vibration, buckling and steady state forced vibration are established by the eigenvalue analysis and expansion of eigenfunction without any trial functions. In addition, the explicit expressions of the characteristic equations, mode functions and steady state response of the nanoplate with two opposite edges that are simply supported or guided supported are obtained. To verify the accuracy and reliability of the present method, numerical results are compared with published solutions and excellent agreement is obtained. Comprehensive benchmark results that consider the nonlocal effect on the dynamic behaviors of rectangular nanoplates are also presented in dimensionless tabular and graphical forms. [ABSTRACT FROM AUTHOR]

• A novel multiscale analysis is developed to assess the buckling behavior of MWCNT-reinforced laminated composites. • Integration of microstructural factors like MWCNT clustering, alignment, and curvature in evaluating composite material performance. • Comprehensive assessment of the influence of moisture and temperature on the buckling performance of MWCNT composites. • Results successfully compared and validated with existing experimental and theoretical data. • Offers significant insights for the design and optimization of MWCNT-reinforced composites. This study introduces a detailed method for analyzing the buckling behavior of laminated composite structures strengthened with multi-walled carbon nanotubes (MWCNTs). We propose a multi-scale analysis that combines analytical and computational techniques to assess the mechanical performance of MWCNT-reinforced composites under combined moisture, temperature, and mechanical stress conditions. The Halpin-Tsai equations are used to calculate the overall stiffness properties of the nano-enhanced matrix, considering factors like MWCNT clustering, alignment, and curvature. Additionally, we incorporate the nanoscopic, size-dependent features of MWCNTs into our model. The Chamis micromechanical formulas are applied to determine the individual elastic properties of the nanocomposite layers, considering the impacts of temperature and moisture. We then explore how variables such as MWCNT content and size, along with temperature and moisture levels, influence the critical buckling load of MWCNT-based laminated composite beams and plates using our multi-scale model. Our results are successfully compared with existing experimental and theoretical data to validate our approach. The developed method offers significant insights for the design and optimization of MWCNT-reinforced composites, potentially benefiting various engineering fields, including aerospace and automotive industries. [ABSTRACT FROM AUTHOR]

In this work, the size-dependent buckling of functionally graded (FG) Bernoulli-Euler beams under non-uniform temperature is analyzed based on the stress-driven nonlocal elasticity and nonlocal heat conduction. By utilizing the variational principle of virtual work, the governing equations and the associated standard boundary conditions are systematically extracted, and the thermal effect, equivalent to the induced thermal load, is explicitly assessed by using the nonlocal heat conduction law. The stress-driven constitutive integral equation is equivalently transformed into a differential form with two non-standard constitutive boundary conditions. By employing the eigenvalue method, the critical buckling loads of the beams with different boundary conditions are obtained. The numerically predicted results reveal that the growth of the nonlocal parameter leads to a consistently strengthening effect on the dimensionless critical buckling loads for all boundary cases. Additionally, the effects of the influential factors pertinent to the nonlocal heat conduction on the buckling behavior are carefully examined. [ABSTRACT FROM AUTHOR]

Many microorganisms propel themselves through complex media by deforming their flagella. The beat is thought to emerge from interactions between forces of the surrounding fluid, the passive elastic response from deformations of the flagellum and active forces from internal molecular motors. The beat varies in response to changes in the fluid rheology, including elasticity, but there are limited data on how systematic changes in elasticity alter the beat. This work analyses a related problem with fixed-strength driving force: the emergence of beating of an elastic planar filament driven by a follower force at the tip of a viscoelastic fluid. This analysis examines how the onset of oscillations depends on the strength of the force and viscoelastic parameters. Compared to a Newtonian fluid, it takes more force to induce the instability in viscoelastic fluids, and the frequency of the oscillation is higher. The linear analysis predicts that the frequency increases with the fluid relaxation time. Using numerical simulations, the model predictions are compared with experimental data on frequency changes in the bi-flagellated alga Chlamydomonas reinhardtii. The model shows the same trends in response to changes in both fluid viscosity and Deborah number and thus provides a possible mechanistic understanding of the experimental observations.

In this paper, we give a targeted review of the state of the art in the study of planar elastic beams in large deformations, also in the presence of geometric nonlinearities. The main scope of this work is to present the different methods of analysis available for describing the possible equilibrium forms and the motions of elastic beams. For the sake of completeness, we start by giving an overview of the nonlinear theories introduced for approaching this argument and then we account for the variational principles and deformation energies introduced for modelling beams undergoing large deformations and displacements. We then consider different kinds of loads treated in the literature and the corresponding induced beam deformations. We conclude by accounting for the available analysis for stability and some considerations about problems where live loads are applied, as well as by describing some relevant numerical methods of use in the applications we have in mind. The selection criterion for the reviewed papers is dictated by the need to study large deformations and the dynamics of pantographic sheets. (Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc R Soc A 2016; 472(2185): 20150790), dell'Isola et al. (Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Z Angew Math Phys 2015; 66(6): 3473–3498), Turco et al. (Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Z Angew Math Phys 2016; 67(4): 1–28)]. [ABSTRACT FROM AUTHOR]

Numerical simulation is used to study the influence of corrosion damage in stiffened plates focusing on elastic buckling strength. Three-dimensional specta are used to simulate geometries of corroded surfaces and finite element method is employed for computing Euler stress of stiffened plates. The influence of corrosion patterns, amount of corrosion loss and roughness of surface are investigated. Ratio of Euler stress of corroded stiffened plate over Euler stress of un-corroded stiffened plate is used to characterize the effects of corrosion on reduction of buckling strength. Results show that reduction of buckling strength is very sensitive to the amount of corrosion loss and roughness of surface, but less sensitive to the location of corroded region. The potential for decrease in buckling strength as a consequence of corrosion is found to depend on the dominant buckling mode. Residual buckling strength is reduced by as much as 12% for the interaction of plate-web-torsional buckling mode, and by 2% for column buckling. [ABSTRACT FROM AUTHOR]

An analytical approach is presented for the accurate definition of lower and upper bounds for the pull-in voltage and tip displacement of a micro- or nanocantilever beam subject to compressive axial load, electrostatic actuation and intermolecular surface forces. The problem is formulated as a nonlinear two-point boundary value problem and has been transformed into an equivalent nonlinear integral equation. Initially, new analytical estimates are found for the beam deflection, which are then employed for assessing novel and accurate bounds from both sides for the pull-in parameters, taking into account for the effects of the compressive axial load. The analytical predictions are found to closely agree with the numerical results provided by the shooting method. The effects of surface elasticity and residual stresses, which are of significant importance when the physical dimensions of structures descend to nanosize, are also included in the proposed approach. [ABSTRACT FROM AUTHOR]

The existing theories of finite-time stability depend on a prescribed bound on initial disturbances and a prescribed threshold for allowable responses. It remains a challenge to identify the critical value of loading parameter for finite time instability observed in experiments without the need of specifying any prescribed threshold for allowable responses. Based on an energy balance analysis of a simple dynamic system, this paper proposes a general criterion for finite time stability which indicates that finite time stability of a linear dynamic system with constant coefficients during a given time interval [0, t ] is guaranteed provided the product of its maximum growth rate (determined by the maximum eigen-root p >0) and the duration t does not exceed 2, i.e., p t <2. The proposed criterion ( p t =2) is applied to several problems of impacted buckling of elastic columns: (i) an elastic column impacted by a striking mass, (ii) longitudinal impact of an elastic column on a rigid wall, and (iii) an elastic column compressed at a constant speed ('Hoff problem'), in which the time-varying axial force is replaced approximately by its average value over the time duration. Comparison of critical parameters predicted by the proposed criterion with available experimental and simulation data shows that the proposed criterion is in robust reasonable agreement with the known data, which suggests that the proposed simple criterion ( p t =2) can be used to estimate critical parameters for finite time stability of dynamic systems governed by linear equations with constant coefficients. [ABSTRACT FROM AUTHOR]

Buckling of centrosymmetric anisotropic beams is studied within strain gradient theory. First, the three dimensional anisotropic gradient elasticity theory is outlined. Then the dimension of the three dimensional theory is reduced, resulting in Timoshenko beam as well as Euler–Bernoulli beam theories. The governing differential equations together with the consistent (classical and non-classical) boundary conditions are derived for centrosymmetric anisotropic beams through a variational approach. By considering von Kármán nonlinear strains, the geometric nonlinearity is taken into account. The obtained nonlinear formulation can be used to study the postbuckling configuration. The analysis of size effect on anisotropic beam structures is missing in the literature so far, while the present model allows one to characterize the size effect on the buckling of the centrosymmetric anisotropic micro- and nano-scale beam structures such as micropillars. As a specific case, the governing buckling equation is obtained for the more practical case of orthotropic beams. Finally, the buckling loads for orthotropic simply supported Timoshenko and Euler–Bernoulli beams as well as a clamped Euler–Bernoulli beam are obtained analytically and the effect of the internal length scale parameters on the buckling load is depicted. [ABSTRACT FROM AUTHOR]