Synchronous/Asynchronous Buckling of Double-Layered Microplate Systems北大核心CSCD

采用修正的偶应力理论和双变量高阶剪切变形理论,发展了层间填充弹性介质的双层微板系统在面内压缩荷载作用下的屈曲模型.基于Euler-Lagrange方程推导了系统屈曲的控制微分方程,运用Navier法获得了上下层均为四边简支时系统同步/异步屈曲的解析解.通过数值算例讨论了系统各参数对其屈曲特性的影响.结果表明：系统的异步屈曲特性依赖于材料尺度参数、长宽比和弹性介质模量,而同步屈曲特性仅依赖于前两项,并且异步屈曲荷载高于同步屈曲荷载;弹性介质的Pasternak模量较之于Winkler模量对系统的屈曲特性影响更显著.     

Local buckling design problem for topology optimized truss girders

The paper presents practical applying of structural topology optimization algorithm based on the formulation of optimal design with compliance minimization. Problem was presented and discussed for several 2D examples. Analysis of local buckling problem was made for various results obtained by the algorithm and approximated into the truss structures. Real bridge load was considered for girders. Topology based optimal girders were compared to typical truss structure. Relation between rods thicknesses was considered. Stress level including buckling in compressed members was examined. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

弹性介质中受轴向冲击载荷作用的圆柱壳的屈曲临界载荷计算分析

建立并求解了弹性介质中圆柱壳的径向位移控制方程,考虑边界条件及相容条件,得到了应力波传播及反射过程中圆柱壳的动力屈曲分叉条件.通过计算得到了不同时间段屈曲临界载荷与应力波波阵面到达圆柱壳的位置、弹性介质的刚度、壳体未嵌入弹性介质部分的长度与总长之比的关系.数值计算结果表明,弹性介质中的圆柱壳发生轴对称屈曲和非轴对称屈曲趋势一致;嵌入弹性介质部分越深、弹性介质刚度越大圆柱壳越难屈曲;屈曲临界载荷随着弹性介质刚度的增大经历了增长缓慢、增长迅速以及增长较慢3个阶段;应力波反射前波阵面通过分界面后,屈曲仅发生在应力波传播区域,反射波波阵面通过分界面前,临界载荷较小时屈曲先发生在反射端部,随着轴向阶数增大,屈曲覆盖整个圆柱壳区域,反射波波阵面通过分界面后,壳体发生的屈曲始终覆盖整个圆柱壳区域.     

Design of planar articulated mechanisms using branch and bound

This paper considers an optimization model and a solution method for the design of two-dimensional mechanical mechanisms. The mechanism design problem is modeled as a nonconvex mixed integer program which allows the optimal topology and geometry of the mechanism to be determined simultaneously. The underlying mechanical analysis model is based on a truss representation allowing for large displacements. For mechanisms undergoing large displacements elastic stability is of major concern. We derive conditions, modeled by nonlinear matrix inequalities, which guarantee that a stable equilibrium is found and that buckling is prevented. The feasible set of the design problem is described by nonlinear differentiable and non-differentiable constraints as well as nonlinear matrix inequalities.To solve the mechanism design problem a branch and bound method based on convex relaxations is developed. To guarantee convergence of the method, two different types of convex relaxations are derived. The relaxations are strengthened by adding valid inequalities to the feasible set and by solving bound contraction sub-problems. Encouraging computational results indicate that the branch and bound method can reliably solve mechanism design problems of realistic size to global optimality.     

Optimal design of multi-purpose structures

The paper presents the optimal (maximum transverse stiffness) design of an elastic, simply supported member of given volume that is to serve as a beam or as a column at different times during its design life. The optimal design can be interpreted in two ways. It is the design that has the maximum Euler buckling load in column action, subject to a prescribed maximum deflection in beam action under a uniformly distributed load; it is also the design that has the least deflection at midspan under a uniformly distributed load, subject to a lower limit on its buckling load in column action. The effectiveness of the optimal design is judged by comparing it with a prismatic bar of the same volume.The author is grateful to Professor W. Prager for suggesting several improvements to an earlier version of the paper.     

Optimal design of multi-purpose beam-columns

The paper deals with the optimal design of an elastic pinended member of given volume that is to serve as a beam for a part of its design life and as a column for the rest. The optimal design can be interpreted in two ways. Firstly, it is the design that has the maximum Euler buckling load in column action, subject to a prescribed maximum deflection as a beam. Secondly, it is the design that has the least deflection as a beam under a midspan concentrated load, subject to a minimum permissible Euler buckling load in column action. The effectiveness of the optimal design is judged by comparing it with a prismatic bar of the same volume.     

阶梯柱屈曲的改进Fourier级数分析

该文对阶梯柱的弹性屈曲问题进行了研究。首先基于改进Fourier级数法采用局部坐标逐段建立阶梯柱的位移函数表达式,然后由带约束的势能变分原理得到含屈曲荷载的线性方程组,利用线性方程组有非零解的条件把问题转化为矩阵特征值问题得到临界载荷,最后讨论方法中的参数取值,并把结果与已有文献和有限元的结果比较,从而验证方法的精度。所提模型在阶梯柱的两端和变截面处引入横向弹簧和旋转弹簧,通过改变弹簧的刚度值模拟不同的边界。所提方法在工程设计中能比较精确地确定各种弹性边界条件下阶梯柱的临界载荷。     

Elastic–plastic buckling of annular circular plate with linear material hardening

The contribution treats the buckling problem of a circular annular plate where the unstable state and the buckling process occurs when the stress state in the most loaded points of the plate is already in the elastic–plastic region. It is supposed that the plate buckles axisymmetrically (m = 0) or nonaxisymmetrically with m > 0, m ∈ ℕ waves in the circumferential direction. Using the equilibrium method, the critical buckling loads are calculated. The results show the influence of the material hardening coefficient 0 ≤ f ≤ 1 on the buckling load. The comparison between calculations with the consideration of flow and deformation theory of plasticity is given.     

弹性直杆中一类动力屈曲问题的渐近解

对于弹性杆受刚性块轴向撞击的动力屈曲问题而言,由于轴向载荷形式较为复杂,问题将归结为关于非线性偏微分方程组解的讨论,至今仍未能得到一个理论上的解析解,为此,讨论了有限长理想弹性直杆的此类动力屈曲问题,采用小参数的摄动展开和变分法,成功地得到了这一问题的一个理论上的近似解,并给出了相应的算例,从中得到了一些有益的结论.     

Exploration of the encased nanocomposites functionally graded porous arches: Nonlinear analysis and stability behavior

This paper explores the nonlinear stability mechanism of the functionally graded porous (FGP) arch reinforced by graphene nanocomposites. Both the pores and the nanocomposites are distributed symmetrically to the mid-surface of the arch but not uniformly in the cross-section so that the bending stiffness can be best improved. The arch is confined in an elastic medium with a radially-pointed concentrated load at the crown position. The confinement of the medium results in a symmetrical deformed shape of the arch, which can be described by an admissible displacement function. Associated with the thin-walled arch theory and the principle of minimum potential energy, analytical predictions are obtained to express the critical buckling load, as well as the hoop force and bending moment. Subsequently, a numerical model is developed to simulate the medium and the arch in ABAQUS software. By introducing the modified arc-length method, the equilibrium paths of the encased arch are traced. After comparison in terms of the critical buckling load and the equilibrium paths, it is found the numerical results are in good accordance with the analytical solutions. Finally, particular attention is paid to the parameters that may impact the buckling load, such as the porosity coefficient, the weight fraction, the central angle, the geometry of the Graphene platelets (GPLs) et al.     

Size-dependent buckling analysis of functionally graded third-order shear deformable microbeams including thermal environment effect

Presented herein is the prediction of buckling behavior of size-dependent microbeams made of functionally graded materials (FGMs) including thermal environment effect. To this purpose, strain gradient elasticity theory is incorporated into the classical third-order shear deformation beam theory to develop a non-classical beam model which contains three additional internal material length scale parameters to consider the effects of size dependencies. The higher-order governing differential equations are derived on the basis of Hamilton’s principle. Afterward, the size-dependent differential equations and related boundary conditions are discretized along with commonly used end supports by employing generalized differential quadrature (GDQ) method. A parametric study is carried out to demonstrate the influences of the dimensionless length scale parameter, material property gradient index, temperature change, length-to-thickness aspect ratio and end supports on the buckling characteristics of FGM microbeams. It is revealed that temperature change plays more important role in the buckling behavior of FGM microbeams with higher values of dimensionless length scale parameter.     

Stability of hydrostatic-pressured FGM thick rings with material nonlinearity

Buckling behaviors of elastoplastic ceramic/metallic functionally graded material (FGM) rings are investigated by using the first order shear deformation theory. The hydrostatic-pressured rings are assumed to be in both the plane-stress case and the plane-strain case, which lead respectively to a uniaxial and a biaxial elastoplastic stress states in prebuckling stage. A uniform strain hypothesis helps to deal with the elastoplastic stress states. By introducing in the graded material properties, the constitutive model of FGMs is formulated under the framework of J₂ deformation theory. By considering the kinetic relations of von-Kárman type and employing the principle of virtual displacement, the equilibrium equations and the buckling governing equations of FGM circular rings are formulated, and the analytical solution of the anisotropic rings is obtained. Finally, the elastoplastic buckling problem is numerically solved through a semi-analytical method, which is proposed to seek the real circumferential strain of FGM rings at the buckling point and determinate the elastoplastic buckling critical hydrostatic pressure. The effects of the inhomogeneous and geometrical parameters on the buckling critical load and the position of the elastoplastic interface are discussed. Results show that, in both the plane-stress and the plane-strain cases, the elastoplastic critical loads are generally lower than their elastic counterparts due to material flow, and the plane-strain critical load is generally larger than the plane-stress one. The elastoplastic critical load does not always decrease monotonously with the increase of the inhomogeneous parameters, which is quite different from their elastic counterparts.     

Buckling of stepped composite columns

The stability of elastic columns subjected to axial pressure is studied. An elastic multistepped column with rectangular cross sections are considered assuming that at the re-entrant corners of the column stable cracks are located. The influence of a crack on the loss of stability of the column is described by means of local flexibility and the function of compliance coupled with the stress intensity factor, which is known from the linear elastic fracture mechanics. A column with a single step is studied in more detail. The influence of crack location and length on the buckling load is assessed numerically. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 1, pp. 87–100, January–February, 2006.     

Stability analysis of graphene based laminated composite sheets under non-uniform inplane loading by nonlocal elasticity

Size dependent buckling of composite laminates made of isotropic graphene layers interlaid with bonding agents is considered. Nonlocal theory of elasticity is used in the buckling analysis to reflect the size scale effects on the critical buckling loads which is discussed in detail. The method is capable of predicting the relative buckling modes for non-uniform inplane loading applied through the thickness of the laminate. All modes of buckling in which the layers may displace together or opposite one another are investigated to study their scale dependent effects. Displacement or load controls are implemented through independent parameters as constraints to form special combination of buckling modes. Each graphene sheet is considered as a Kirchhoff plate model. The interlaid bonding agent is laterally treated as Winkler elastic foundation between graphene layers while neglecting their other load carrying capacities. Various numerical results are obtained reflecting the nonlocality effects. It is observed that in cases of higher load ratios and simpler buckling modes, the effect of nonlocality tends to drastically increase. The results of simpler examples studied are verified by another reference.     

Nonlocal effect on axially compressed buckling of triple-walled carbon nanotubes under temperature field

This paper studies the small scale effect on the buckling behaviors of triple-walled carbon nanotubes (TWCNTs) with the initial axial stress under the temperature field. The TWCNTs are modeled as three elastic shells coupled together through vdW interaction between different layers. Buckling governing equations of CNTs are firstly formulated on the basis of nonlocal elastic theory and the small scale effect on CNTs buckling results with the change of temperature are then achieved. The results show that the critical buckling load is dependent on the temperature, scale parameter and wavenumber. Some conclusions are drawn that small scale effect will arise gradually with the increases of wavenumber, and the temperature can influence the ratio between the nonlocal buckling load and the corresponding local load. Furthermore, with or without effects of nonlocal considered, the same results is obtained that the axial buckling load increases as the value of temperature increases at low and room temperature condition, while at high temperature condition the axial buckling load decreases as the value of temperature increases.     

An analytical spectral stiffness method for buckling of rectangular plates on Winkler foundation subject to general boundary conditions

An analytical spectral stiffness method is proposed for the efficient and accurate buckling analysis of rectangular plates on Winkler foundation subject to general boundary conditions (BCs). The method combines the advantages of superposition method, stiffness-based method and the Wittrick–Williams algorithm. First, exact general solutions of the governing differential equation (GDE) of plate buckling considering both elastic foundation and biaxial loading is derived by using a modified Fourier series. The superposition of such general solutions satisfy the GDE exactly and BCs approximately, which guarantees the rapid convergence and high accuracy. Then, based on the exact general solution, the spectral stiffness matrix which relates the coefficients of plate generalized displacement BCs and force BCs is symbolically developed. As a result, arbitrary BCs can be prescribed straightforwardly in the stiffness-based model. As an efficient and reliable solution technique, the Wittrick–Williams algorithm with the J₀ problem resolved is applied to obtain the critical buckling solutions. The accuracy and efficiency of the method are verified by comparing with other methods. Benchmark buckling solutions are provided for plates with all possible boundary conditions. Also, dependence of various factors such as foundation stiffness, load combinations and aspect ratio on the buckling behaviors are investigated.     

Calibration of Eringen's small length scale coefficient for buckling circular and annular plates via Hencky bar-net model

This paper is focused on the modeling of circular and annular graphene sheets via Hencky bar-net model (HBM¹) and calibrating the Eringen's small length scale coefficient e₀ in Eringen's nonlocal theory. The buckling solutions of circular and annular graphene sheets based on Eringen's nonlocal continuum plate theory are first obtained. On the other hand, HBM is developed to model the same structure from the discrete view. HBM is a grid system comprising rigid bars and arcs connected by frictionless hinges with elastic rotational and torsional springs. By regarding the length of straight segments in HBM equal to the characteristic length of Eringen's nonlocal model (ENM²) and matching their solutions, the Eringen's small length scale coefficient e₀ is calibrated. It is found that for circular graphene sheet, e₀ = 0.258 for clamped edge and e₀ = 0.300 for simply supported edge. For annular graphene sheet, e₀ is dependent on the inner to outer radius ratio χ and boundary conditions. The scale coefficient e₀ takes 0.307–0.367 for clamped edges while 0.219–0.290 for simply supported edges with χ varying from 0.2 to 0.8. Another finding is that the graphene sheet will buckle with a very small load when its dimension is large, regardless of models adopted. However for small dimensions, ENM and HBM predict lower buckling loads than the classical local model because the scale effect is more obvious.     

Buckling localization in a non‐elastic rotating disk

Buckling localization of a rotating disk made of elastic‐perfectly plastic material is investigated using stress‐rate formulation of the stability boundary‐value problem. The phenomenon of plastic buckling localization and its analogy with elastic buckling localization is discussed. For a thin rotating disk, it is shown that buckling develops at a speed lower than one at which the disk passes to fully plastic state, or in other words, before the limit load has been attained. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

2D elastic analysis of the sandwich panel buckling problem: benchmark solutions and accurate finite element formulations

This paper is concerned with the elastic stability of a sandwich beam panel using classical elasticity. An exact solution for the buckling problem of a sandwich panel (wide beam) in uniaxial compression is presented. Various formulations that correspond to the use of different pairs of energetically conjugate stress and strain measures for the infinitesimal elastic stability of the sandwich panel are discussed. Results from the present two-dimensional analyses to predict the global and local buckling of a sandwich panel are compared with previous theoretical and experimental results. A new finite element formulation for the bifurcation buckling problem is also introduced. In this new formulation, terms that influence the buckling load, which have been omitted in popular commercial codes are pointed out and their significance in influencing the buckling load is identified. The formulation and results presented here can be used as a benchmark solution to establish the accuracy of numerical methods for computing the buckling behavior of thick, orthotropic solids.     

2D elastic analysis of the sandwich panel buckling problem: benchmark solutions and accurate finite element formulations

This paper is concerned with the elastic stability of a sandwich beam panel using classical elasticity. An exact solution for the buckling problem of a sandwich panel (wide beam) in uniaxial compression is presented. Various formulations that correspond to the use of different pairs of energetically conjugate stress and strain measures for the infinitesimal elastic stability of the sandwich panel are discussed. Results from the present two-dimensional analyses to predict the global and local buckling of a sandwich panel are compared with previous theoretical and experimental results. A new finite element formulation for the bifurcation buckling problem is also introduced. In this new formulation, terms that influence the buckling load, which have been omitted in popular commercial codes are pointed out and their significance in influencing the buckling load is identified. The formulation and results presented here can be used as a benchmark solution to establish the accuracy of numerical methods for computing the buckling behavior of thick, orthotropic solids.