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Abstract: The mechanical buckling of curved beams made of functionally graded materials is studies in this paper. The equilibrium and stability equations of curved beams under mechanical loads are derived. Using proper approximate functions for the displacement components, the stability equations are employed to obtain the related eigenvalues associated with the buckling load of the curved beam. Closed-form solutions are obtained for mechanical buckling of curved beams with doubly symmetric cross section subjected to uniform distributed radial load and pure bending moment. The results are validated with the known data in the literature for beams with isotropic materials. [Copyright &y& Elsevier]

Abstract: A generalized layer-wise stochastic finite element formulation is developed for the buckling analysis of both homogeneous and laminated plates with random material properties. The pre-buckled stresses are considered in the derivation of geometric stiffness matrix and the effect of variation in these stresses on the mean and coefficient of variation of buckling strength is studied. The mean buckling strength of plates under uniform stress assumption exactly matches with those reported in the literature. However, it is shown that the actual mean buckling strength of plates can be significantly different based on the pre-buckled stress analysis which depends on boundary constraints, principal material directions, aspect and thickness ratios of plates. The statistics of buckling strength is determined using a Taylor series expansion based mean centered first order perturbation technique. The stochastic finite element solutions obtained using layer-wise plate theory is also validated with analytical solutions presented in this paper. Parametric studies are conducted for different aspect ratios, ply orientations and boundary conditions. [Copyright &y& Elsevier]

Abstract: A finite element model is presented to study the static and dynamic stability of a pre-twisted Timoshenko beam having asymmetric aerofoil cross-section subjected to an axial periodic force. The change in the effects of coupling due to centre of flexure distance from the centroid and the shear coefficient on the stability due to the pre-twist angle is considered. It is concluded from the results that the pre-twist angle has an influence on the effects of coupling and the shear coefficient. [Copyright &y& Elsevier]

Abstract: An overview, unification and review of the mathematical models, stability criteria and critical parameters for two types of structures, patched plates and layered shells, subjected to uniform temperature change are presented. Results of numerical simulations based on analytical solutions are presented and critical behavior is reviewed and compared. In each case, a transverse loading parameter is seen to force the system. For both types of structure, the loading parameter is dependent on both the temperature and the membrane force, and a critical membrane force and critical temperature are identified. When the corresponding critical temperature is achieved, both types of structures considered are seen to exhibit sling-shot buckling, an instability in which the structure dynamically slings from equilibrium configurations associated with deflections in one direction to those associated with deflections in the opposite sense. Conclusions are drawn and explanations are given as to the physical reasons for this phenomenon. [Copyright &y& Elsevier]

Abstract: The buckling characteristics of sandwich plates having laminated stiff layers are studied for different types of partial edge loadings using a refined plate theory. With this plate theory, the through thickness variation of transverse shear stresses is represented by piecewise parabolic functions where the continuity of these stresses is satisfied at the layer interfaces by taking jumps in the transverse shear strains at the interfaces. The transverse shear stresses free condition at the plate top and bottom surfaces is also satisfied. It is quite interesting to note that this plate model having all these refined features requires unknown parameters only at the reference plane. To have a generality in the present analysis, finite element technique is adopted and it is carried out with newly developed triangular element, as existing finite elements cannot accommodate this plate model. So far, no solution exists in the literature for the problem of sandwich plate subjected to partial edge loading. The present analysis is first validated for the case of an isotropic plate subjected to partial edge compression and then it is extended to analyze sandwich plates. Few results are presented. [Copyright &y& Elsevier]

Abstract: The collapse behaviour of pin-ended steel columns with unequal flanges is investigated using a modified version of the Shanley model. Particular attention is given to the effect of the geometrical imperfection size on the behaviour of columns having a range of flange area ratios and slenderness values. Columns which are initially straight are dealt with in a companion paper (The collapse of steel model columns with unequal flanges. I. Perfect. International Journal of Mechanical Sciences, in press, doi:10.1016/j.ijmecsci.2004.09.005). Compressive yield of the smaller flange is confirmed as an important source of failure for the column. However, tensile yield of this flange can also cause failure, particularly at relatively high slenderness and/or imperfection size. Failure caused by yielding of the smaller flange, whether compressive or tensile, is often characterised by violent load-shedding, and high imperfection-sensitivity, and this is most noticeable at intermediate slenderness. A graphical method is suggested for predicting the maximum load, and for showing the degree of imperfection sensitivity. [Copyright &y& Elsevier]

Abstract: The paper is concerned with the collapse behaviour of initially straight pin-ended steel columns having unequal flanges. A modified Shanley model column with unequal flanges is presented and analysed, together with the corresponding modified Calladine construction. Each column flange is assumed to have a uniform state of stress and strain, and to behave elastically until fully yielding in compression or tension—thus local buckling is not included. However, column buckling is fully allowed for, and the complete collapse behaviour of various model column designs is investigated, the parameters varied being the normalised column slenderness, and the flange area ratio. The behaviour of the model columns is shown to agree well with that of corresponding real columns, i.e., columns of uniform cross-section that deform at all points within the volume. Compressive yield of the smaller flange is an expected source of failure, but the results show that tensile yield of the smaller flange can be equally important. Violent collapse behaviour is shown to be possible whether the smaller flange yields in compression or tension. This is particularly so at intermediate slenderness. The results apply most directly to unsymmetrical I-sections, but are also relevant to T-sections, and to wide stiffened compression panels. The Calladine construction greatly elucidates behaviour. [Copyright &y& Elsevier]

An elastic buckling theory is developed for thin-walled arches. Using the principle of minimum total potential energy derives the governing differential equations. An explicit and clear approximation of the curvature effect is made in the derivation process. Closed form solutions are obtained for arches subjected to equal and opposite end moments (uniform bending) and to uniformly distributed radial loads (uniform compression). Also, closed form solutions for the torsional buckling moment of braced arches in uniform bending and for the torsional buckling load of braced arches in uniform compression are obtained. The solutions are compared with previous theoretical results. [Copyright &y& Elsevier]

This paper examines the buckling and collapse of cylindrical shells under axial load with one end radially and tangentially fixed, with varying axial fixity, and the other end free. The bifurcation loads are found for elastic cylinders, while collapse loads are found for both elastic and elastic-perfectly plastic cylinders. The varying axial restraint is applied in the form of linear springs. The eigenvalue buckling loads are calculated with conditions matching those of a classical analysis. Bifurcation loads are shown to be a function of the axial restraint; as the axial restraint is increased, the bifurcation load increases dramatically, until it reaches that of a semi-infinite, open ended cylinder. A non-dimensional form of the axial spring stiffness is proposed, and shown to be applicable across a range of geometries.The collapse load and imperfection sensitivity of cylinders with the boundary conditions examined here is also found to be a function of the axial restraint. Cylinders with low axial restraint are shown to be imperfection insensitive, with collapse loads above, or close to, the bifurcation load. As the amount of axial restraint increases, the collapse behaviour displays a degree of imperfection sensitivity associated with more usual boundary conditions. [Copyright &y& Elsevier]

The effect of the prebuckling in-plane deformations on the elastic flexural-torsional buckling of laterally fixed circular arches is studied in this paper. The finite strains and the energy equation for the flexural-torsional buckling of arches have been derived based on an accurate orthogonal rotation matrix. A closed form solution for the elastic flexural-torsional buckling resistance of laterally fixed arches in uniform bending, including the effects of the prebuckling deformations, is obtained. It is found that the notion that the prebuckling deformations increase the flexural-torsional buckling moment of an arch or of a beam is not necessarily correct for a laterally fixed arch or beam in uniform bending, in deference to a laterally pinned arch. When a laterally fixed arch is subjected to positive uniform bending, the effects of the prebuckling deformations decrease the buckling moment, and the reduction of the buckling moment increases with an increase of the included angle and of the out-of-plane slenderness ratio of the arch. When a laterally fixed arch is subjected to negative uniform bending, the effects of the prebuckling deformations decrease the absolute value of its buckling moment when the included angle is very small, but increase the absolute value of the buckling moment when the included angle exceeds a certain value. The increase in the absolute value of the buckling moment increases with an increase of the included angle and of the out-of-plane slenderness ratio of the arch. When the ratio of the out-of-plane to the in-plane second moments of area of the cross-section is not small, both the reduction of the buckling moment of a laterally fixed arch in positive uniform bending and the increase of the buckling moment of a laterally fixed arch in negative uniform bending, are substantial. [Copyright &y& Elsevier]

Based on the mathematical formulations for predicting forming limits induced by buckling, wrinkling and bursting of free-expansion tube hydroforming, a theoretical “Process Window Diagram” (PWD) is proposed and established in this paper. The theory developed in the first part of the present work was formulated within the context of free-expansion tube hydroforming with both combined internal pressure and end feeding. The PWD is designed to provide a quick assessment of part producibility for tube hydroforming. The predicted PWD is validated against experimental results conducted for 6260-T4 60×2×320 (mm) aluminum tubes. An optimal loading path is also proposed in the PWD with an attempt to define the ideal forming process for aluminum tube hydroforming. Parametric studies show that the PWD has a strong dependency on tube geometry, material property and process parameters. To the authors’ knowledge, this is the first attempt that a PWD is being formulated theoretically. Such a concept can be advantageous in deriving design solutions and determining optimal process parameters for tube hydroforming processes. [Copyright &y& Elsevier]

Three possible failure modes have been identified in tube hydroforming: buckling, wrinkling and bursting. A general theoretical framework is proposed for analyzing these failure modes as an elastoplastic bifurcation problem. This framework enables advanced yield criteria and various strain-hardening laws to be readily incorporated into the analysis. The effect of plastic deformation on the geometric instability in tube hydroforming, such as global buckling, axisymmetric wrinkling and asymmetric wrinkling, is precisely treated by using the exact plane stress moduli tensor. A mathematical formulation for predicting the localized condition for bursting failure is established herein. Furthermore, the critical conditions governing the onset of buckling, axisymmetric wrinkling and asymmetric wrinkling are derived in closed-form expressions for the critical axial compressive stresses. Closed-form solutions for the critical stress are developed based on Neale–Hutchinson''s constitutive equation and an assumed deformation theory of plasticity. It is demonstrated that the onset of asymmetric wrinkling always requires a higher critical axial compressive stress than the axisymmetric one under the context of tube hydroforming with applied internal pressure and hence the asymmetric wrinkling mode can be excluded in the analysis of tube hydroforming. Parametric studies show that buckling and axisymmetric wrinkling are strongly dependent on geometric parameters such as t0/r0 and r0/ℓ0, and that axisymmetric wrinkling is the predominant mode for short tubes while global buckling occurs for long slender tubes. [Copyright &y& Elsevier]

A new efficient coupled one-dimensional (1D) geometrically nonlinear zigzag theory is developed for buckling analysis of hybrid piezoelectric beams, under electromechanical loads. The potential field is approximated layerwise as piecewise linear. The deflection is approximated to account for the normal strain due to electric field. The axial displacement is approximated as a combination of a global third-order variation and layerwise linear variation. It is expressed in terms of three primary displacement variables and a set of electric potential variables by enforcing exactly the conditions of zero transverse shear stress at the top and bottom and the conditions of its continuity at the layer interfaces. The governing coupled nonlinear field equations and boundary conditions are derived using a variational principle. Analytical solutions for buckling of simply supported beams under electromechanical loads are presented. Comparisons with the exact 2D piezoelasticity solution establish that the present zigzag theory is very accurate for buckling analysis. [Copyright &y& Elsevier]

The large deflection equation of a shallow spherical shell under uniformly distributed transverse loads is established in this paper with consideration of effects of transverse shear deformation on flexural deformation. Using an updated iteration method, an analytical solution for nonlinear stability of a shallow spherical shell is obtained. Formulae for estimating the critical buckling loads are presented for two types of boundary conditions. Discussions on the influences of the geometric and physical parameters on the critical buckling loads are given. [Copyright &y& Elsevier]

The finite element method is applied to study the static and dynamic stability of an aerofoil cross-section rotating blade subjected to an axial periodic force. The effects of coupling due to the center of flexure (shear center) distance from the centroid, rotational speed, stagger angle and disk radius on the stability are considered. Critical load parameters are calculated and dynamic instability regions of the blade with different reference values are illustrated graphically. The numerical results indicate that the coupling effect is very important in the third and fourth modes depending on dY distance. The increase in stagger angle makes the blade less stable. However, the increase in rotational speed and disk radius make the blade more stable. [Copyright &y& Elsevier]

This paper investigates the buckling behavior of a symmetrically laminated composite rectangular plate composed of AS4 graphite fibers and 3501-6 epoxy resin under parabolic variation of axial loads. The plate is assumed to be general anisotropy and symmetric about its mid-plane. Two loading conditions are considered. They are (a) an axial load that varies parabolically with respect to the plate longitudinal direction, and (b) an axial load that varies parabolically with respect to the plate transverse direction. Analytical solutions of buckling load and mode shape are obtained based on classical laminated plate theory and the Rayleigh–Ritz method. The influences of plate aspect ratio and fiber orientations have been investigated. Some numerical results from the present study are compared with published results and good agreement is found. [Copyright &y& Elsevier]

In this paper, the moving least-squares differential quadrature (MLSDQ) method is applied to the bending and buckling analyses of moderately thick symmetric laminates based on the first-order shear deformation theory (FSDT). The transverse deflection and two rotations of the laminate are independently assumed with the moving least-squares (MLS) approximation. The weighting coefficients used in the MLSDQ approximation are obtained through a fast computation of the MLS shape functions and their partial derivatives. Numerical examples are illustrated to study the accuracy, stability and convergence of the MLSDQ method. The typical displacements, stresses and critical buckling loads of various laminated plates are presented and compared with the analytical values. Effects of support size, order of the complete basis functions and node irregularity on the numerical accuracy are investigated. [Copyright &y& Elsevier]

Exact solutions are presented for the free vibration and buckling of rectangular plates having two opposite edges (x=0 and a) simply supported and the other two (y=0 and b) clamped, with the simply supported edges subjected to a linearly varying normal stress σx=−N0[1−α(y/b)]/h, where h is the plate thickness. By assuming the transverse displacement (w) to vary as sin(mπx/a), the governing partial differential equation of motion is reduced to an ordinary differential equation in y with variable coefficients, for which an exact solution is obtained as a power series (the method of Frobenius). Applying the clamped boundary conditions at y=0 and b yields the frequency determinant. Buckling loads arise as the frequencies approach zero. A careful study of the convergence of the power series is made. Buckling loads are determined for loading parameters α=0,0.5,1,1.5,2, for which α=2 is a pure in-plane bending moment. Comparisons are made with published buckling loads for α=0,1,2 obtained by the method of integration of the differential equation (α=0) or the method of energy (α=1,2). Novel results are presented for the free vibration frequencies of rectangular plates with aspect ratios a/b=0.5,1,2 subjected to three types of loadings (α=0,1,2), with load intensities N0/Ncr=0,0.5,0.8,0.95,1, where Ncr is the critical buckling load of the plate. Contour plots of buckling and free vibration mode shapes are also shown. [Copyright &y& Elsevier]

The classical theory of buckling of axially loaded thin cylindrical shells predicts that the buckling stress is directly proportional to the thickness t, other things being equal. But empirical data show clearly that the buckling stress is actually proportional to t1.5, other things being equal. As is well known, there is wide scatter in the buckling-stress data, going from one half to twice the mean value for a given ratio R/t. Current theories of shell buckling explain the low buckling stress—in comparison with the classical—and the experimental scatter in terms of “imperfection-sensitive”, non-linear behaviour. But those theories always take the classical analysis of an ideal, perfect shell as their point of reference.Our present principal aim is to explain the observed t1.5 law. So far as we know, no previous attack has been made on this particular aspect of thin-shell buckling. Our work is thus breaking new ground, and we shall deliberately avoid taking the classical analysis as our starting point.We first point out that experiments on self-weight buckling of open-topped cylindrical shells agree well with the mean experimental data mentioned above; and then we associate those results with a well-defined post-buckling “plateau” in load/deflection space, that is revealed by finite-element studies. This plateau is linked with the appearance of a characteristic “dimple” of a mainly inextensional character in the deformed shell wall. A somewhat similar post-buckling dimple is also found by quite separate finite-element studies when a thin cylindrical shell is loaded axially at an edge by a localised force; and it turns out that such a dimple grows under a more-or-less constant force that is proportional to t2.5, other things being equal.This 2.5-power law can be explained by analogy with the inversion of a thin spherical shell by an inward-directed force. Thus, the deformation of such a shell is generally inextensional except for a narrow “knuckle” or boundary layer in which the combined local elastic energy of bending and stretching is proportional to t2.5, other things being equal. Similarly, the modes of deformation in the post-buckling dimples in a cylindrical shell are practically independent of thickness, except in the highly deformed boundary-layer regions which separate the inextensionally distorted portions of the shell. These ideas lead in turn to an explanation of the t1.5 law for the post-buckling stress of open-topped cylindrical shells loaded by their own weight.We attribute the absence of experimental scatter in the self-weight buckling of open-topped cylindrical shells to the statical determinacy of the situation, which allows a post-buckling dimple to grow at a well-defined “plateau load”. Conversely, the large experimental scatter in tests on cylinders with closed ends may be attributed to the lack of statical determinacy there.Our paper contains several arguments that are not mathematically water-tight, in contrast to many reports in the field of mechanics of structures. We plead that the problem which we have tackled is so difficult that the only way forward is one of “over-simplification”. We hope that our work will be judged not with respect to its absence of mathematical precision, but by the light which it sheds upon the problem under investigation. [Copyright &y& Elsevier]

A meshless approach based on the reproducing kernel particle method is developed for the flexural, free vibration and buckling analysis of laminated composite plates. In this approach, the first-order shear deformation theory (FSDT) is employed and the displacement shape functions are constructed using the reproducing kernel approximation satisfying the consistency conditions. The essential boundary conditions are enforced by a singular kernel method. Numerical examples involving various boundary conditions are solved to demonstrate the validity of the proposed method. Comparison of results with the exact and other known solutions in the literature suggests that the meshless approach yields an effective solution method for laminated composite plates. [Copyright &y& Elsevier]

The process of rotational symmetric tube bulging with inside pressure and axial compression enables the standard tubes to be formed into different rotational symmetric hollow parts in such a way that their central part is expanded into a desired shape while the ends remain unchanged. The superposition of axial compression contributes to a more favorable forming stress state, which is reflected in larger forming limits and smaller wall thinning in the widened area. The problems characterizing the process are a limited range of compression stability and difficulties met when establishing and optimizing the technological parameters of the process whose course cannot be defined in an analytical way. Based on a physical model of the forming process a numerical model was built. Using ABAQUS code the model was simulated over the entire stress/forming region. The comparison of the computer simulated forming process with the experimentally obtained results showed that the model was highly accurate. Finally, the paper studies the influences of particular parameters on the stability of the process, showing on a practical example how it is possible to achieve tube bulging without wall thinning. [Copyright &y& Elsevier]

The natural vibrations and critical loads of foundation beams embedded in a soil simulated with two elastic parameters through the Winkler–Pasternak (WP) model are analysed. General end supports of the beam are considered by introducing elastic constraints to transversal displacements and rotations. The solution is tackled by means of a direct variational methodology previously developed by the authors who named it as whole element method. The solution is stated by means of extended trigonometric series. This method gives rise to theoretically exact natural frequencies and critical loads. A particular behaviour arises from the analysis of the lateral soil influence. It is found that the boundary conditions of the beam are influenced by the soil at the left and right sides of the beam. The possible alternatives are that the soil be cut or dragged by the non-fixed ends of the beam. In the standard WP model, the lateral soil influence is not considered. Natural frequencies and critical load numerical values are reported for beams and piles elastically supported and for various soil parameters. The results are found with arbitrary precision depending on the number of terms taken in the series. Some unexpected modes and eigenvalues are found when the different alternatives are studied. It should be noted that this special behaviour is present only when the Pasternak contribution is taken into account. [Copyright &y& Elsevier]

• Symplectic superposition method for buckling of side-cracked plates is developed. • New analytic buckling solutions of side-cracked rectangular plates are obtained. • Comprehensive benchmark numerical and graphical results are presented. • The present method provides a rational approach to exploring analytic solutions. Exploring new analytic buckling solutions of cracked plates is of much importance for providing benchmark results and implementing preliminary structural designs based on explicit parametric analysis and optimization. However, the governing higher-order partial differential equations as well as the geometric discontinuity across a crack essentially complicate the problems, make it more intractable to seeking analytic solutions. This paper presents a first attempt to extend an up-to-date symplectic superposition method to linear buckling of side-cracked rectangular thin plates. The problems are introduced into the Hamiltonian system, and a side-cracked plate is then divided into several sub-plates that are analytically solved by the symplectic superposition method, where the symplectic eigenvalue problems are formulated, followed by the symplectic eigen expansion. Some elaborated mechanical quantities are imposed on the edges of each sub-plate, with multiple sets of constants determined by the actual boundary conditions of the cracked plate, free edge conditions along the crack, and interfacial continuity conditions among the sub-plates. The final analytic solution of a side-cracked plate is obtained by integration of the solutions of the sub-plates. Comprehensive benchmark results are tabulated and plotted for buckling loads and mode shapes of typical plates with a side crack from a simply supported edge, a clamped edge, or a free edge. The crack length effect is also investigated by the analytic solutions obtained. Due to the rigorous mathematical derivations without predetermination of solution forms, the present method provides a rational approach to exploring more analytic solutions. Image, graphical abstract [ABSTRACT FROM AUTHOR]

• The discrete stiffness parameters are employed to yield the theoretical optimal ply orientation that corresponds to the maximum critical buckling load of simply supported orthotropic plates at a layer level, then, the optimal stacking sequence are obtained with the derived optimal ply orientation and a sign vector. • Two laminate optimal design problems are addressed: the critical buckling load with fixed thickness is maximized, and the thickness with buckling constraint is minimized. • The derived maximum critical buckling load can be extended to the symmetric and antisymmetric angle-ply laminated plates with small coupling stiffness B 16 and B 26. • The theoretical optimal ply orientations exhibiting a range of load ratios and aspect ratios are presented. The buckling characteristic of rectangular orthotropic plates under axial compression with simply supported boundary has been extensively studied. To enhance the structural efficiency, the stacking sequence of laminate should be optimized to maximize the critical buckling load. In this study, the discrete stiffness parameters are employed to yield the theoretical optimal ply orientation corresponding to the maximum critical buckling load at a layer level, and the buckling mode shapes are calculated analytically. Subsequently, the derivation is performed based on optimal ply orientation to derive the optimal stacking sequence of the laminate, in which a sign vector is adopted to minimize the bending-twisting coupling effects. As a result, two laminate optimal design problems are solved: maximizing critical buckling load with fixed thickness, as well as minimizing the thickness with buckling constraint. Two numerical examples are adopted to verify the yielded optimal solutions. Lastly, the theoretical optimal ply orientations exhibiting different load ratios and aspect ratios are presented. Image, graphical abstract [ABSTRACT FROM AUTHOR]

• Incorporating surface effect can significantly alter the estimated buckling and vibration results of single crystalline nanobeams. • Besides the beam geometric parameters, elements of metal and crystallographic directions will also significantly influence the results. • When nanobeams are thin and with significant surface effects, the generic 3rd order beam theory gives more accurate results due to the inclusion of the non-vanishing shear stress effect at the interface between the surface layer and the bulk of the beams. • If nanobeam thickness is more than 80nm and with less significant surface effects, there is negligible difference of the buckling and vibration results obtained either using the 1st order, the Reddy 3rd order or the generic 3rd order beam theories. A novel approach for the vibration and buckling analysis of nanobeams with surface stress effects is presented in this paper. The material model is derived in which the bulk core of a beam is assumed to be a single crystalline material and have certain crystallographic directions. The beams are assumed to be covered by atomic layers with the same crystallographic directions on the top and bottom surfaces. A generic third order shear deformable beam theory is employed to derive the total energy functional of the beams. Minimizing the total potential energy functional by applying the p-Ritz method, the eigenvalue equations for the buckling and vibration of nanobeams with surface stress effects are obtained. The buckling and vibration behaviours of the beams considering the surface stress effects are discussed in detail and are compared with the results obtained from the first order beam theory and the Reddy's third order beam theory. It is found that in certain cases, both the first order and the Reddy's third order beam theories may not be able to capture the surface effect accurately on the vibration and buckling behaviours of the nanobeams. The effects of surface stresses on the buckling and vibration behaviours for nanobeams made of Ni single crystals obtained by the generic third order beam theory showed opposite trends to the ones obtained by the first order or the Reddy's third beam theory. Image, graphical abstract [ABSTRACT FROM AUTHOR]

• Detailed buckling analysis of thin annular sheets subjected to partial shrinkage is presented. • In-plane hoop compression due to the competition between shrinkage and non-shrinkage zones can induce buckling despite free boundary conditions at the edges of the sheet. • A simple scaling argument demonstrates that minimised (critical) buckling strain is obtained when the shrinkage area corresponds to the non-shrinkage area. • The critical buckling mode can switch from a saddle to ripple mode when Young's modulus in the shrinkage zone is approximately ten times larger than that of the non-shrinkage, depending on the shrinkage size and radius ratio of the annuli. This paper reports a detailed buckling analysis of a free-standing annular sheet subjected to partial shrinkage as one of the strategies of morphing structures, which has received considerable attention lately. The linear buckling problem can be decomposed into in-plane and out-of-plane deformations. The in-plane (Lamé) problem is first analysed to determine the stress distribution and show that in-plane hoop compression due to the competition between shrinkage and non-shrinkage zones can induce buckling despite free boundary conditions at the edges of the sheet. Subsequently, buckling analysis is performed to numerically determine buckling strains, to explore the effect of material and geometrical parameters on their buckling thresholds. A simple scaling argument is used to demonstrate that minimised (critical) buckling strain to the shrinkage radius is obtained when the shrinkage area corresponds to the non-shrinkage area. The results demonstrate that critical buckling mode can switch from a saddle to ripple mode when Young's modulus in the shrinkage zone is approximately ten times that of the non-shrinkage zone, depending on the shrinkage size and radius ratio of the annuli. [ABSTRACT FROM AUTHOR]

• A unified Timoshenko model with transition parameter is adopted in analyses. • Analytical solutions are obtained explicitly in terms of elementary functions. • Inherent relations of the buckling loads for two classical beam models are extracted. • Frequency equations are derived via the method of separation of variable. • Effects of transition parameter are quantified on deformations of beams. This paper aims to investigate the bending, buckling and free vibration problems of an axially loaded Timoshenko beam in a systematic manner. A recently-developed unified model is adopted, where a transition parameter characterizing the direction of the axial force during the deformation is introduced. For beams with various boundary conditions, closed-form expressions of deflections and critical buckling loadings are obtained in an analytical fashion and frequency equations are derived. It is highlighted that the inherent relations of the buckling loadings for the Timoshenko and Euler-Bernoulli beams are explicitly extracted. Numerical calculations are carried out to check the validity of the present solutions and to quantify the effects of the transition parameter. Significant effects of the transition parameter are revealed on the bending, buckling and free vibration of the axially loaded beam. The present analysis is of significance to axially-loaded beam-like mechanical system, especially for determination of the direction of axially force. [ABSTRACT FROM AUTHOR]

• A new method for the determination of mixed-mode buckling loads of composite beams. • Results are of excellent accuracy when compared to finite element simulations. • Negligible computational effort in comparison with finite element simulations. • Ready-to-use efficient analysis method for practical engineering purposes. This paper presents a novel semi-analytical approach for the determination of the local buckling loads of moderately thick-walled composite laminated doubly-symmetric I-beams under uniaxial compression. The composite laminates that make up the flanges and webs of the I-beam are assumed to exhibit symmetric orthotropic layups and are further assumed to be moderately thick so that a higher order laminate theory, namely Reddy's Third Order Shear Deformation Theory, is employed. Contrary to the commonly used approach of discrete plate analysis for local beam buckling calculations, i.e. the separate consideration of flanges and webs, the analysis method that is presented in this paper considers the interaction between webs and flanges by using series expansions for all buckling degrees of freedom and a subsequent assembly by employing adequate continuity conditions at those locations where adjacent segments of the cross-sections intersect. Such a coupled analysis will be called mixed-mode buckling in the course of this paper. The current analysis method uses the Ritz method in conjunction with the principle of minimum elastic potential of the buckled beam, and it will be shown that the method converges rapidly with a very low number of shape functions in the employed series expansions so that it works with superior numerical efficiency when compared to full-scale finite element computations, however with a comparable results accuracy which makes the present approach especially attractive for any practical application purpose where the local buckling behavior of moderately thick-walled composite laminated beams needs to be considered. Image, graphical abstract [ABSTRACT FROM AUTHOR]

• Performed in-planar shear test on ultra-thin stainless steel sheet. • Buckling is suppressed by adding supporting on two-side surfaces. • Evaluated the strain path at the notch region experimentally and numerically. • Optimised the notch geometries to extend the shear strain path. Bipolar plates are essential components in proton exchange membrane (PEM) fuel cells for electric vehicles. Micro-formed thin (thickness <0.15 mm) stainless steel bipolar plates have advantages over conventional CNC-machined graphite plates in terms of weight and cost, but the geometry of the final part shape is limited due to the fracture limits of the material. The further optimisation of micro-forming processes, therefore, requires the development of reliable fracture models, with fracture locus covering a wide range of stress states, including shear fracture at zero (or close to zero) stress triaxiality conditions. Few studies have investigated shear tests on thin specimens, as it is challenging to maintain a stable shear condition during the test. There exists a twofold problem that: (a) the sheet buckles prematurely before the failure strain is reached; (b) plane strain fracture initiates prematurely from the edge of the gauge area. In this study, in-planar shear tests are performed on 0.1 mm-thick stainless steel sheet. To achieve fracture initiation in shear, an anti-buckling device was developed and the shear test sample shape was optimised. The shape optimization process is based on finite element analysis (FEA) and verification with experimental shear test trials. The optimised shape has a minimum value of a goal function that encompasses the Lode parameter, stress triaxiality and the ratio of the centre to edge equivalent plastic strain. The experimental trials indicate that the optimised sample shape, in combination with the clamping device, enables testing thin stainless steel for fracture limits in shear. The test routine can be used to calibrate ductile fracture models. [ABSTRACT FROM AUTHOR]

• In-plane response of CFST arches under thermal and mechanical loading is studied. • Analytical solutions for elastic anti-symmetric buckling loads are derived. • Novel numerical method proposed for inelastic pre-buckling analysis. • Inelastic buckling strength investigated using Finite-Element Analysis. • Thermal loading is found to reduce buckling strength of CFST arches. This paper analytically and numerically investigates the pre-buckling response and in-plane stability boundaries of circular concrete-filled steel tubular (CFST) arches subjected to combined thermal and mechanical loading. The governing non-linear equations of equilibrium are obtained using energy methods and both elastic and inelastic material behaviour is considered. A novel mechanically derived non-discretisation numerical method is proposed for the pre-buckling analysis. The stress-strain relation of the confining steel tube is described using a bi-linear plasticity model, and an inelastic material model is adopted for the concrete core which considers the effects of confinement and transient thermal strain. The result is a system of first-order differential equations which can be numerically solved with known boundary conditions including fixed ends, pinned ends or crowned-pinned cases. Closed-form solutions are presented for the elastic anti-symmetric bifurcation loads, whilst the inelastic anti-symmetric buckling strength was studied using finite element (FE) analysis. The FE model is verified by comparison to the derived analytical and numerical models which show a high level of agreement. Additionally, a sensitivity analysis is conducted which explores the influence of the constitutive material law for the concrete core and contact model for the steel-concrete interface on critical buckling loads. [ABSTRACT FROM AUTHOR]