On the integral equation method for buckling loads

An initial value method for the integral equation of the column is presented for determining the buckling load of columns. The differential equation of the column is reduced to a Fredholm integral equation. An initial value problem is derived for this integral equation, which is reduced to a set of ordinary differential equations with prescribed initial conditions in order to find the Fredholm resolvent. The singularities of the resolvent occur at the eigenvalues. Integration of the equations proceeds until the integrals become excessively large, indicating that a critical load has been reached. To check this method, numerical results are given for two examples, for which the critical load is well known. One is the Euler load of a simply supported beam, and the other case is the buckling load of a cantilever beam under its own weight. The advantage of this initial value method is that it can be applied easily to solve other nonlinear problems for which the critical loads are unknown. This approach will be illustrated in future papers.     

Static analysis of functionally graded beams using higher order shear deformation theory

Displacement field based on higher order shear deformation theory is implemented to study the static behavior of functionally graded metal–ceramic (FGM) beams under ambient temperature. FGM beams with variation of volume fraction of metal or ceramic based on power law exponent are considered. Using the principle of stationary potential energy, the finite element form of static equilibrium equation for FGM beam is presented. Two stiffness matrices are thus derived so that one among them will reflect the influence of rotation of the normal and the other shear rotation. Numerical results on the transverse deflection, axial and shear stresses in a moderately thick FGM beam under uniform distributed load for clamped–clamped and simply supported boundary conditions are discussed in depth. The effect of power law exponent for various combination of metal–ceramic FGM beam on the deflection and stresses are also commented. The studies reveal that, depending on whether the loading is on the ceramic rich face or metal rich face of the beam, the static deflection and the static stresses in the beam do not remain the same.     

Elastic buckling of a thin eccentric circular annulus

The contribution treats the elastic buckling of a thin eccentric circular annulus which is on the inner and on the outer boundaries subjected to uniform and constant pressure or tensile loads. The inner and outer boundary are simply supported. To determine the plane stress state and the critical outer load, all equations are expressed with complex variables in the complex plane (z), and conformally mapped into a new complex plane (ζ). The energy method is used for the determination of a critical outer load at which the buckling process appears.     

Variant of an Analytical Shear Model for the Stress-Strain State of Heterogeneous Composite Beams

A nonclassical analytical model for the stress-strain state of composite beams with account of shear strains is suggested. It is assumed that the beam is piecewise heterogeneous across its height. Normal and tangential loads operate on its upper and lower surfaces and on interfaces. The model describes the distribution of tangential displacements across the thickness of plies by a third-degree polynomial. The corresponding system of differential equations is obtained by the variational method and contains two equations. The first one is an analog of the equation of classical theory of beams for deflections, and the second one is an analog of the equation of the theory for the bending moment from the generalized load. The solutions to test problems are compared with three-dimensional solutions and with experimental results for simply supported and clamped beams of different composite structure. An applied engineering problem is solved for a multispan statically indeterminate beam.     

A general nonlocal nonlinear model for buckling of nanobeams

This study presents a unified model for the nonlocal response of nanobeams in buckling and postbuckling states. The formulation is suitable for the classical Euler–Bernoulli, first-order Timoshenko, and higher-order shear deformation beam theories. The small-scale effect is modeled according to the nonlocal elasticity theory of Eringen. The equations of equilibrium are obtained using the principle of virtual work. The stress resultants are developed taking into account the nonlocal effect. Analytical solutions for the critical buckling load and the amplitude of the static nonlinear response in the postbuckling state are obtained. It is found out that as the nonlocal parameter increases, the critical buckling load reduces and the amplitude of buckling increases. Numerical results showing variation of the critical buckling load and the amplitude of buckling with the nonlocal parameter and the length-to-height ratio for simply supported and clamped–clamped nanobeams are presented.     

多元函数极值原理在某力学问题中的应用

利用力学中的最小势能原理建立了任意有限个移动荷载在简支梁上平行移动时,简支梁的挠度方程解析表达式;利用多元函数的极值原理给出了确定简支梁绝对最大挠度最不利位置的解析判别公式.     

基于修正偶应力理论的Timoshenko微梁模型和尺寸效应研究

基于修正偶应力理论,将Timoshenko微梁的应力、偶应力、应变、曲率等基本变量,描述为位移分量偏导数的表达式.根据最小势能原理,推导了决定Timoshenko微梁位移场的位移场控微分方程.利用级数法求解了任意载荷作用下Timoshenko简支微梁的位移场控微分方程,得到了反映尺寸效应的挠度、转角及应力的偶应力理论解.通过对承受余弦分布载荷Timoshenko简支微梁的数值计算,研究了Timoshenko微梁的挠度、转角和应力的尺寸效应,分析了Poisson比对Timoshenko微梁力学行为及其尺寸效应的影响.结果表明:当截面高度与材料特征长度的比值小于5时,Timoshenko微梁的刚度和强度均随着截面高度的减小而显著提高,表现出明显的尺寸效应;当截面高度与材料特征长度的比值大于10时,Timoshenko微梁的刚度与强度均趋于稳定,尺寸效应可以忽略;材料Poisson比是影响Timoshenko微梁力学行为及尺寸效应的重要因素,Poisson比越大Timoshenko微梁刚度和强度的尺寸效应越显著.该文建立的Timoshenko微梁模型,能有效描述Timoshenko微梁的力学行为及尺寸效应,可为微电子机械系统（MEMS）中的微结构设计与分析提供理论基础和技术参考.     

薄球壳在均布外压与温度耦合作用下的热屈曲研究

从张量方法推导出的轴对称薄球壳屈曲方程出发,推导出在均布外压与温度耦合作用下用位移表示的薄球壳热屈曲方程;应用虚功原理建立薄球壳屈曲最小势能泛函;进一步用Ritz(里兹)法分析了周边简支的半球壳的3种热屈曲问题.得到了: 1) 温度不超过屈曲临界温度值时,均布外压的临界载荷;2) 均布外压载荷为0时,屈曲临界温度值;3) 均布外压载荷不超过临界载荷时,屈曲临界温度值.     

A free boundary problem and stability for the nonlinear beam

The stability bound for the classical nonlinear Euler beam is determined in the case that its deflection is limited by an obstacle parallel to the plane of the beam. Let a clamped or simply supported beam be axially compressed by a force P > P₀, where P₀ denotes the critical load. So far only a linear theory has been applied to analyze the stability of the solutions in contact with the obstacle and the jumping to a different state. Utilizing a free boundary problem formulation we analytically as well as numerically answer these questions for the nonlinear beam.     

The equivalence of the refined theory and the decomposition theorem of rectangular beams

This paper presents the decomposition theorem of rectangular beams and indicates that the general state of stress of beams can be decomposed into two parts: the interior state and the Papkovich–Fadle state (shortened form the P–F state). The refined theory of beams is derived by using Papkovich–Neuber solution (shortened form the P–N solution) and Lur’e method without ad hoc assumptions. It is shown that the displacements and stresses of the beam can be represented by the angle of rotation and the deflection of the neutral surface. Based on the refined beam theory, the exact equations for the beam without transverse surface loadings are derived and consist of two governing differential equations: the fourth-order equation and the transcendental equation. It is then proved that the refined beam theory and the decomposition beam theorem are equivalent, i.e., the fourth-order equation and the transcendental equation are equivalent to the interior state and the P–F state, respectively.     

Nonlinear analysis of a parametrically excited beam with intermediate support by using Multi-dimensional incremental harmonic balance method

In this paper, a nonlinear Euler-Bernoulli beam under a concentrated harmonic excitation with intermediate nonlinear support is investigated. Continuous expression for the kinetic energy, potential energy and dissipation function are constructed. An energy method based on the Lagrange equation combined with the Galerkin truncation is used for discretizing the governing equation. The Multi-dimensional incremental harmonic balance method (MIHBM) is derived, and the comparisons between the numerical results and the approximate analytical solutions based on the MIHBM verify the excellent accuracy of the MIHBM. The steady state dynamic of the beam is investigated by MIHBM. In order to investigate the energy transmission and understand the vibration response of the Euler-Bernoulli beam, the effects of the key parameters on the dynamic behaviors are studied and discussed, individually. The results show that the amplitude-frequency curves exhibits softening nonlinear behavior in the super-harmonic resonance region, and near resonant region the hardening nonlinear behavior is observed depending on the different parameters. Nonlinear dynamic analysis, such as bifurcation, 3-D frequency spectrum, waveform, frequency spectrum, phase diagram and Poincaré map, are also presented in order to study the influences of the key parameters on the vibration behaviors for the beam in a more accurate manner. In addition, the path to chaotic motion is observed to be through a sequence of the periodic motion and quasi-periodic motion.     

Two-dimensional thermoelastic analysis of beams with variable thickness subjected to thermo-mechanical loads

Two-dimensional thermoelastic analysis for simply supported beams with variable thickness and subjected to thermo-mechanical loads is investigated. An approximate analytical method is proposed. Firstly, the heat conduction equation is analytically solved to obtain the temperature distributions for two kinds of boundary conditions at the beam ends, which are the harmonic series with unknown coefficients. Then the two-dimensional equilibrium differential equations are analytically solved to obtain the displacement component series with unknown coefficients and the stress component series is obtained. The unknown coefficients in the temperature series and the stress component series are approximately determined by using the upper surface and lower surface conditions of the beam. With the proposed procedure, the solutions satisfy the governing differential equations, the loading conditions, and the simply supported end conditions. The proposed solution method shows a good convergence and the results agree well with those obtained from the commercial finite element software ANSYS. Several examples are used to demonstrate the effectiveness of the proposed solution method. The simultaneous effects of temperature change and applied mechanical load on the behavior of the beam are examined.     

一类不可压广义neo-Hookean球体的空穴分岔问题的定性研究

研究了一类不可压的广义neo-Hookean材料组成的球体的空穴分岔问题,该类材料可以看作是带有径向摄动的均匀各向同性不可压的neo-Hookean材料,得到了球体内部空穴生成的条件．与均匀各向同性的neo-Hookean球体的情况相比,证明了当摄动参数属于某些区域时,从平凡解局部向左分岔的空穴分岔解上存在一个二次转向分岔点,空穴生成时的临界载荷会比无摄动的材料的临界载荷小．用奇点理论证明了,空穴分岔方程在临界点附近等价于具有单边约束条件的正规形．用最小势能原理分别讨论了空穴分岔解的稳定性和实际稳定的平衡状态．     

A new Bernoulli–Euler beam model incorporating microstructure and surface energy effects

A new Bernoulli–Euler beam model is developed using a modified couple stress theory and a surface elasticity theory. A variational formulation based on the principle of minimum total potential energy is employed, which leads to the simultaneous determination of the equilibrium equation and complete boundary conditions for a Bernoulli–Euler beam. The new model contains a material length scale parameter accounting for the microstructure effect in the bulk of the beam and three surface elasticity constants describing the mechanical behavior of the beam surface layer. The inclusion of these additional material constants enables the new model to capture the microstructure- and surface energy-dependent size effect. In addition, Poisson’s effect is incorporated in the current model, unlike existing beam models. The new beam model includes the models considering only the microstructure dependence or the surface energy effect as special cases. The current model reduces to the classical Bernoulli–Euler beam model when the microstructure dependence, surface energy, and Poisson’s effect are all suppressed. To demonstrate the new model, a cantilever beam problem is solved by directly applying the general formulas derived. Numerical results reveal that the beam deflection predicted by the new model is smaller than that by the classical beam model. Also, it is found that the difference between the deflections predicted by the two models is very significant when the beam thickness is small but is diminishing with the increase of the beam thickness.     

Dynamic stability of a porous cylindrical shell

This paper is devoted to a closed cylindrical shell made of a porous-cellular material. The mechanical properties vary continuously on the thickness of a shell. The mechanical model of porosity is as described as presented by Magnucki, Stasiewicz. A shell is simply supported on edges. On the ground of assumed displacement functions the deformation of shell is defined. The displacement field of any cross section and linear geometrical and physical relationships are assumed in cylindrical coordinate system. The components of deformation and stress state were found. Using the Hamilton's principle the system of differential equations of dynamic stability is obtained. The forms of unknown functions are assumed and the system of a differential equations is reduced to a simple ordinary equation of dynamic stability of shell (Mathieu's equation). The derived equation are used for solving a problem of dynamic stability of porous-cellular shell with intensity of load directed in generators of shell. The critical loads are derived for a family of porous shells. The unstable space of family porous shells is found. The influence a coefficient of porosity on the stability regions in Figures is presented. The results obtained for porous shell are compared to a homogeneous isotropic cylindrical shell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Structural Crack Identification Based on the Variational Mode Decomposition北大核心CSCD

In order to enrich the bridge damage detection method and further improve the accuracy of bridge damage identification, a detection method for simply supported beams with cracks under dynamic loads was proposed not based on the complete finite element model. Under the premise of not blocking traffic, the method only needs to analyze and deal with the acceleration responses of the simply supported beam span, which reduces the mounting, dismounting and maintenance of sensors in practical engineering. At the same time, based on the model, an analytical formula of the acceleration at the midspan of the simply supported cracked beam was derived. Based on the theoretical derivation, the instantaneous energy and the mean energy difference were constructed through the variational mode decomposition and the Hilbert transform, and these 2 crack identification indexes were used to effectively identify small cracks with a crack depth ratio of only 5%. Then the influences of different wheel loads, environmental noises and damage degrees on detection results were studied. The results show that: ① the instantaneous frequency has a better recognition effect for crack positions; ② the mean energy difference is sensitive to crack depth ratio δ and the wheel load magnitude; ③ this method has strong noise robustness. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Buckling mode localization in rib-stiffened plates with misplaced stiffeners – Kantorovich approach

Buckling mode localization in rib-stiffened plates with randomly misplaced stiffeners is studied in this paper. The method of Kantorovich on reducing a partial differential equation to a system of ordinary differential equations is employed to obtain the deflection surface of the rib-stiffened plates under axial compressive load. The edges of the plates normal to the stiffeners can be either simply supported or clamped. The solutions of the deflection surface are then expressed in the form of transfer matrices. The expressions of the solutions obtained for the case of one edge simply supported and one edge clamped and the case of two edges clamped are similar to those for the case of two edges simply supported. When the two edges are simply supported, the method of Kantorovich yields the exact results. Localization factors, which characterize the average exponential rates of growth or decay of amplitudes of deflection, are determined using the method of transfer matrix. The method of Kantorovich is a general approximate method, which is applicable for various support conditions.     

柔性约束下压杆的一些稳定和不稳定的临界状态

研究了一端固定、一端弹簧约束滑动固定的压杆在Euler临界载荷作用下的稳定性.将系统的势能表示为转角的泛函,将扰动量展开成Fourier级数,将势能的二阶变分表示成一个二次型,得到在临界状态下势能的二阶变分半正定,并求得临界载荷与屈曲模态.进一步研究临界状态下高阶变分的正定性,包括四阶和六阶变分的正定性.结果表明,与刚性约束不同的是,柔性约束压杆临界状态的稳定性与约束的刚度有关,有稳定与不稳定之分,并给出了临界状态是稳定和不稳定的情况下柔性约束相对刚度的范围.     

考虑阻尼的状态向量方程和层合板的振动分析

基于考虑弹性体粘滞阻尼的修正后的Hellinger-Reissner (H-R)变分原理,推导了相应的状态向量方程．结合精细积分法和Muller法为四边简支矩形层合板的简谐振动分析提出了新的方法．依据线性阻尼振动理论,简要地给出了复合材料层合板欠阻尼、临界阻尼和过阻尼3种自由运动的通解公式．通过数值实例研究了粘滞阻尼对复合材料层合板振动的影响．丰富了状态向量方程的理论体系和应用领域．