Free vibration and buckling of tapered columns made of axially functionally graded materials

This paper presents a unified model to analyze the free vibration and buckling of axially functionally graded Euler-Bernoulli columns subjected to an axial compressive force. The material properties vary linearly along the longitudinal direction, and column with circular and square cross sections is linearly tapered. The governing differential equations of the problem are derived and solved using the direct integral method combined with the determinant search technique. The computed results are compared with those reported in the literature and obtained from the finite element software ADINA. Numerical examples for natural frequency, buckling load and their corresponding mode shapes are given to highlight the effects of modular ratio, taper ratio and cross sectional shape as well as the end condition.     

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.     

Stability and vibration analysis of axially-loaded shear beam-columns carrying elastically restrained mass

Classical shear beams only consider the deflection resulting from sliding of parallel cross-sections, and do not consider the effect of rotation of cross-sections. Adopting the Kausel beam theory where cross-sectional rotation is considered, this article studies stability and free vibration of axially-loaded shear beams using Engesser’s and Haringx’s approaches. For attached mass at elastically supported ends, we present a unified analytical approach for obtaining a characteristic equation. By setting natural frequencies to be zero in this equation, critical buckling load can be determined. The resulting frequency equation reduces to the classical one when cross-sections do not rotate. The mode shapes at free vibration and buckling are given. The frequency equations for shear beam-columns with special free/pinned/clamped ends and carrying concentrated mass at the end can be obtained from the present. The influences of elastic restraint coefficients, axial loads and moment of inertia on the natural frequencies and buckling loads are expounded. It is found that the Engesser theory is superior to the Haringx theory.     

变截面箱形薄壁立柱屈曲荷载的近似分析解

变截面箱形薄壁立柱弯扭屈曲的三个控制方程是二阶或四阶变系数的常微分方程,很难用解析的方法求解。本文用多项式来近似截面的几何特性和微分方程的某些系数,用能量原理和伽辽金法分别导出了计算这种立柱弯曲和扭转屈曲荷载的近似公式,用数值算例来验证了所给解答的正确性。本文的计算结果为论证变截面箱形薄壁立柱的稳定性提供了依据。本文具有实用价值。     

Free vibration and stability of tapered Euler–Bernoulli beams made of axially functionally graded materials

The free vibration and stability of axially functionally graded tapered Euler–Bernoulli beams are studied through solving the governing differential equations of motion. Observing the fact that the conventional differential transform method (DTM) does not necessarily converge to satisfactory results, a new approach based on DTM called differential transform element method (DTEM) is introduced which considerably improves the convergence rate of the method. In addition to DTEM, differential quadrature element method of lowest-order (DQEL) is used to solve the governing differential equation, as well. Carrying out several numerical examples, the competency of DQEL and DTEM in determination of free longitudinal and free transverse frequencies and critical buckling load of tapered Euler–Bernoulli beams made of axially functionally graded materials is verified.     

Size dependent buckling analysis of functionally graded micro beams based on modified couple stress theory

Buckling analysis of functionally graded micro beams based on modified couple stress theory is presented. Three different beam theories, i.e. classical, first and third order shear deformation beam theories, are considered to study the effect of shear deformations. To present a profound insight on the effect of boundary conditions, beams with hinged-hinged, clamped–clamped and clamped–hinged ends are studied. Governing equations and boundary conditions are derived using principle of minimum potential energy. Afterwards, generalized differential quadrature (GDQ) method is applied to solve the obtained differential equations. Some numerical results are presented to study the effects of material length scale parameter, beam thickness, Poisson ratio and power index of material distribution on size dependent buckling load. It is observed that buckling loads predicted by modified couple stress theory deviates significantly from classical ones, especially for thin beams. It is shown that size dependency of FG micro beams differs from isotropic homogeneous micro beams as it is a function of power index of material distribution. In addition, the general trend of buckling load with respect to Poisson ratio predicted by the present model differs from classical one.     

Novel closed – form solutions in buckling of inhomogeneous columns under distributed variable loading

In this study we consider buckling of columns with variable stiffness, under axially distributed loading varying polynomially. The objective is to obtain closed – form solutions for the buckling load. The problem is posed in inverse setting: determine the column’s stiffness, so that it has the given, polynomial, buckling mode. Four sets of boundary conditions are investigated. Some perplexing results are obtained, namely, that irrespective of boundary conditions, the critical load of the column is the same; this occurs in conjunction with the fact that the obtained distribution for stiffness is different for each set of boundary conditions.     

Vibration and stability of ring-stiffened Euler–Bernoulli tie-bars

Tie-bars are frequently used in structural and mechanical engineering applications. To satisfy requirements like weight reduction, stability improvement, etc., the tie-bars are stiffened with rings. In this paper a method is developed to calculate the natural frequencies, buckling axial force, etc., of the ring-stiffened tie-bars. The dynamics of the ringed and the unringed portions of the beam are treated separately. The mode shape of the first portion was expressed as the superposition of two functions multiplied by constants. Consideration of continuity of deflection and of slope and compatibility of bending moment and shearing force at the first step enabled the mode shape of the second portion to be expressed as the superposition of two functions but multiplied by the same constants as in the first portion. This procedure was recursively carried out up to the last portion. The frequency equation was then derived from the boundary conditions at the end. Buckling of the tie-bar was considered as the case when one of the natural frequencies is zero. The first three frequency parameters and the first two buckling dimensionless axial forces are tabulated for tie-bars stiffened with various number of rings and for various combinations of boundary conditions. The calculation procedure can handle any number and any type of ring-stiffeners.     

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.     

Buckling and free vibration analysis of thick rectangular plates resting on elastic foundation using mixed finite element and differential quadrature method

In this article, a combination of the finite element (FE) and differential quadrature (DQ) methods is used to solve the eigenvalue (buckling and free vibration) equations of rectangular thick plates resting on elastic foundations. The elastic foundation is described by the Pasternak (two-parameter) model. The three dimensional, linear and small strain theory of elasticity and energy principle are employed to derive the governing equations. The in-plane domain is discretized using two dimensional finite elements. The spatial derivatives of equations in the thickness direction are discretized in strong-form using DQM. Buckling and free vibration of rectangular thick plates of various thicknesses to width and aspect ratios with Pasternak elastic foundation are investigated using the proposed FE-DQ method. The results obtained by the mixed method have been verified by the few analytical solutions in the literature. It is concluded that the mixed FE-DQ method has good convergancy behavior; and acceptable accuracy can be obtained by the method with a reasonable degrees of freedom.     

Geometrically exact solution of a buckling column with asymmetric boundary conditions

For the symmetrically supported Euler buckling column with both ends hinged the classical stability theory yields simple trigonometric functions as buckling modes, i.e. w(x) = A sin αx. The eigenvalues α are just multiples of π. In comparison, the analysis of the asymmetrically supported Euler buckling column with one end clamped and the other end hinged is more complicated: The buckling modes are a combination of trigonometric functions in form of w(x) = A (sin αx − αx cos (αL)). The eigenvalues follow from a transcendental equation. Applying a geometrically exact theory to the aforementioned Euler buckling problems, a similar relation in the complexity of the analyses will naturally arise. Using, e.g., the elastica model the buckling behavior of the symmetrically supported column is represented by elliptic integrals. However, the determination of the buckling behavior of the asymmetrically supported column turns out to be much more complex and elaborate. This article presents a direct comparison of the symmetrically and asymmetrically supported buckling columns regarding their analyses by means of classical stability theory and by the geometrically exact theory of the elastica. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Static and dynamic buckling modes of spherical shells subjected to external pressure

We investigate the possibilities for simplification of previously proposed refined linearized equations of perturbedmotion to identify, by dynamic method, the buckling mode shapes of isotropic spherical shells undergoing external hydrodynamic pressure. In the analysis of classical flexural buckling shapes of spherical shells, it is shown that preserving of nonconservative parametric terms in governing equations of formulated problem, which are related with loading of the shell with follower pressure practically does not affect the value of critical load and the resulting bucklingmode shapes in shell.     

Torsional buckling and post-buckling of columns made of aluminium alloy

The paper concerns torsional buckling and the initial post-buckling of axially compressed thin-walled aluminium alloy columns with bisymmetrical cross-section. It is assumed that the column material behaviour is described by the Ramberg–Osgood constitutive equation in non-linear elastic range. The stationary total energy principle is used to derive the governing non-linear differential equation. An approximate solution of the equation determined by means of the perturbation approach allows to determine the buckling loads and the initial post-buckling behaviour. Numerical examples dealing with simply supported I-column are presented and the effect of material elastic non-linearity on the critical loads and initial post-buckling behaviour are compared to the linear solution.     

Optimal placement of available sections in structural eigenvalue problems

Basic principles governing the optimum design of a structure of given geometry to resist buckling are restated, expanded, and generalized. The specific case of column buckling is cast in a complementary energy format. This format is used to treat the discrete optimality problem of finding the best location and length of cover plates of given cross-sectional area. Both necessary and sufficient optimality conditions are derived, and an iterative design-analysis sequence is established, whose convergence is proved. A simple numerical example demonstrates the procedure. Although the strict proof of sufficiency is confined to simply supported and cantilever columns, a possible extension to other cases is introduced at the end of the paper.     

Semi-analytical approximations for a class of multi-parameter eigenvalue problems related to tensile buckling

This work complements the first author’s recent asymptotic investigations on a class of boundary-eigenvalue problems for tensile buckling of thin elastic plates. In particular, it is shown here that the approximations for the critical buckling load can be improved by using a modified energy method that relies directly on the asymptotic results derived previously. We also explore a number of additional mathematical features that have an intrinsic interest in the context of multi-parameter eigenvalue problems.     

An efficient mixed methodology for free vibration and buckling analysis of orthotropic rectangular plates

In this paper, a simple and efficient mixed Ritz-differential quadrature (DQ) method is presented for free vibration and buckling analysis of orthotropic rectangular plates. The mixed scheme combines the simplicity of the Ritz method and high accuracy and efficiency of the DQ method. The accuracy of the proposed method is demonstrated by comparing the calculated results with those available in the literature. It is shown that highly accurate results can be obtained using a small number of Ritz terms and DQ sample points. The proposed method is suitable for the problem considered due to its simplicity and potential for further development.     

Instability Paths in the Kirchhoff–Plateau Problem

The Kirchhoff–Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a soap film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. Adopting a variational approach, we define an energy associated with shape deformations of the system and then derive general equilibrium and (linear) stability conditions by considering the first and second variations of the energy functional. We analyze in detail the transition to instability of flat circular configurations, which are ground states for the system in the absence of surface tension, when the latter is progressively increased. Such a theoretical study is particularly useful here, since the many different perturbations that can lead to instability make it challenging to perform an exhaustive experimental investigation. We generalize previous results, since we allow the filament to possess a curved intrinsic shape and also to display anisotropic flexural properties (as happens when the cross section of the filament is noncircular). This is accomplished by using a rod energy which is familiar from the modeling of DNA filaments. We find that the presence of intrinsic curvature is necessary to obtain a first buckling mode which is not purely tangent to the spanning surface. We also elucidate the role of twisting buckling modes, which become relevant in the presence of flexural anisotropy.     

轴向冲击下理想塑性直杆动态屈曲的过应力简化分析模型*

本文在理想塑性直杆的动态屈曲分析中引入应变率效应,得到相应的动力学微分方程,求出了屈曲半波长,临界载荷和屈曲时间的表达式.讨论了应变率效应对杆的塑性动态屈曲的影响.并与文[4]的理论和试验结果作了比较.     

Dynamic analysis of a micro-resonator driven by electrostatic combs

The dynamic response of a micro-resonator driven by electrostatic combs is investigated in this work. The micro-resonator is assumed to consist of eight flexible beams and three rigid bodies. The nonlinear partial differential equations that govern the motions of the flexible beams are obtained, as well as their boundary and matching conditions. The natural matching conditions for the flexible beams are the governing equations for the rigid bodies. The undamped natural frequencies and mode shapes of the linearized model of the micro-resonator are determined, and the orthogonality relation of the undamped global mode shapes is established. The modified Newton iterative method is used to simultaneously solve for the frequency equation and identify repeated natural frequencies that can occur in the micro-resonator and their multiplicities. The Gram-Schmidt orthogonalization method is extended to orthogonalize the mode shapes of the continuous system corresponding to the repeated natural frequencies. The undamped global mode shapes are used to spatially discretize the nonlinear partial differential equations of the micro-resonator. The simulation results show that the geometric nonlinearities of the flexible beams can have a significant effect on the dynamic response of the micro-resonator.     

Finite element buckling analysis of multi-layered graphene sheets on elastic substrate based on nonlocal elasticity theory

Graphene-polymer nano-composites are one of the most applicable engineering nanostructures with superior mechanical properties. In the present study, a finite element (FE) approach based on the size dependent nonlocal elasticity theory is developed for buckling analysis of nano-scaled multi-layered graphene sheets (MLGSs) embedded in polymer matrix. The van der Waals (vdW) interactions between the graphene layers and graphene-polymer are simulated as a set of linear springs using the Lennard-Jones potential model. The governing stability equations for nonlocal classical orthotropic plates together with the weighted residual formulation are employed to explicitly obtain stiffness and buckling matrices for a multi-layered super element of MLGS. The accuracy of the current finite element analysis (FEA) is approved through a comparison with molecular dynamics (MD) and analytical solutions available in the literature. Effects of nonlocal parameter, dimensions, vdW interactions, elastic foundation, mode numbers and boundary conditions on critical in-plane loads are investigated for different types of MLGS. It is found that buckling loads of MLGS are generally of two types namely In-Phase (INPH) and Out-of-Phase (OPH) loads. The INPH loads are independent of interlayer vdW interactions while the OPH loads depend on vdW interactions. It is seen that the decreasing effect of nonlocal parameter on the OPH buckling loads dwindles as the interlayer vdW interactions become stronger. Also, it is found that the small scale and polymer substrate have noticeable effects on the buckling loads of embedded MLGS.