Bending of functionally graded cantilever beam with power-law non-linearity subjected to an end force

A realistic beam structure often exhibits material and geometrical non-linearity, in particular for those made of metals. The mechanical behaviors of a non-linear functionally graded-material (FGM) cantilever beam subjected to an end force are investigated by using large and small deformation theories. Young's modulus is assumed to be depth-dependent. For an FGM beam of power-law hardening, the location of the neutral axis is determined. The effects of depth-dependent Young's modulus and non-linearity parameter on the deflections and rotations of the FGM beams are analyzed. Our results show that different gradient indexes may change the bending stiffness of the beam so that an FGM beam may bear larger applied load than a homogeneous beam when choosing appropriate gradients. Moreover, the bending stress distribution in an FGM beam is completely different from that in a homogeneous beam. The bending stress arrives at the maximum tensile stress at an internal position rather than at the surface. Obtained results are useful in safety design of linear and non-linear beams.     

热载荷作用下梯度多孔材料梁弯曲及过屈曲力学行为的研究

基于Bernoulli-Euler梁理论,引入物理中面解耦了复合材料结构的面内变形与横向弯曲特性,研究了梯度多孔材料矩形截面梁在热载荷作用下的弯曲及过屈曲力学行为．假设沿梁厚度方向材料的性质是连续变化的,利用能量法推导了矩形截面梁的控制微分方程和边界条件,并用打靶法对无量纲化的控制方程进行数值求解．利用计算得到的结果分析了材料的性质、热载荷、边界条件对矩形截面梁非线性力学行为的影响．结果表明,对称材料模型下,固支梁与简支梁均显示出了典型的分支屈曲行为特征,而其临界屈曲热载荷值均会随着孔隙率系数的增加而单调增加．非对称材料模型下,固支梁仍显示出分支屈曲行为特征,但其临界屈曲热载荷不再随着孔隙率系数的变化而单调变化;而对于两端简支梁,发生了弯曲变形,弯曲挠度随载荷的增大而增大．     

纵横载荷作用下功能梯度梁的弯曲和过屈曲

基于一阶非线性梁理论和物理中面概念,导出了纵横向载荷作用下功能梯度材料(FGM)梁非线性弯曲和过屈曲问题的控制方程,并获得了该问题的精确解;据此解研究了梯度材料性质、外载荷、横向剪切变形以及边界条件等因素对功能梯度材料梁非线性力学行为的影响,分析中假设功能梯度材料性质只沿梁厚度方向,并按成分含量的幂指数函数形式变化。结果表明:纵横载荷共同作用下,功能梯度梁的弯曲构形将有无限多个;随着梯度指数的增大,梁的变形减小,临界载荷升高;随着长高比的增大,横向剪切变形的影响减小。     

弹性地基双层梁理论下的混凝土路面力学分析

为建立混凝土路面结构受力分析计算模型,以Winkler弹性地基梁模型为基础,推导出了弹性地基双层梁理论的表达式;给定边界条件,利用MATLAB软件获得了无限长弹性地基梁在集中力作用下的挠度表达式。将混凝土路面结构简化为弹性地基上的双层梁,当车辆荷载作用于混凝土路面时,在集中载荷的作用下,建立了面层与基层的微分平衡方程。应用广义“初参数”法,得到了双层梁位移和应力的解析解。通过算例,对面层及基层的变形和应力进行了分析,结果表明:增大面层、基层的轴惯性矩和地基的弹性常数,可以有效地减少面层和基层的变形量,降低最大应力数值,但抗弯刚度对基层和面层的弯矩受力影响不大。最后将结果与ANSYS分析结果进行了比较,佐证了解的可靠性,研究结果可为混凝土路面结构设计提供依据。     

Large displacement analysis of elastic pyramidal trusses

The paper examines the equilibrium stability problem for a simple class of elastic space trusses in the shape of a regular pyramid. Joints located at the vertices of the base polygon are fixed while the joint at the apex is subjected to a proportionally increasing load acting in either the vertical direction, in the horizontal plane, or along a generic oblique direction. Exact closed-form solutions are derived for each load condition under the common hypotheses of linear material law, small or moderate axial deformation in bars and large nodal displacements. Despite their seeming simplicity, these mechanical systems exhibit a wide variety of post-critical responses, not exhausted by the classical snapping and bifurcation phenomena. In addition to regular primary and secondary branches, the equilibrium paths may include neutral branches, namely branches entirely composed of bifurcation or limit points. Besides their immediate theoretical interest, these branches are particularly difficult to handle by the standard numerical procedures of non-linear analysis, so the given solutions may represent severe benchmark tests.     

Quasi-static response of linear viscoelastic cantilever beams subject to a concentrated harmonic end load

This study evaluates the response of a uniform cantilever beam with a symmetric cross-section fixed at one end, and submitted to a lateral concentrated sinusoidal load at the free extremity. The beam material is assumed to be homogeneous, isotropic and linear viscoelastic. Due to the nature of the loading and the beam slenderness, large displacements are developed but the strains are considered small. Consequently, the mathematical formulation only involves geometrical non-linearity. It is also assumed that the beam is inextensible (neutral axis length is constant) and that inertial forces are negligible, i.e., dynamic effects are insignificant and the system can thus be modeled quasi-statically. The beam is therefore subject to oscillations caused by the sinusoidal time-dependent load, leading to a transient response until the material stabilizes and the system exhibits a periodic response, which can be conveniently described in the frequency domain. The time domain solution of this problem is elaborated by considering the quasi-static response for each time interval. The mathematical equations are presented in dimensional and dimensionless forms, and for the latter case, a numerical solution is generated and several case studies are presented. The problem is governed by a set of non-linear ordinary differential equations encompassing functions of space and time that relate the curvature, rotation angle, bending moment and geometrical coordinates. In this study, an elegant solution is deduced using perturbation theory, yielding a precise steady-state solution in the frequency domain with considerable computational economy. The solutions for both time and frequency domain methods are developed and compared using a case study for a series of dimensionless parameters that influence the response of the system.     

The non-linear bending of a cantilever beam with shear and longitudinal deformations

A non-linear bending theory for beams is constructed which accommodates shear and longitudinal deformations. Using the theory, an analytical solution for the cantilever beam subject to a compressive load (the elastica) and a series solution for the horizontal cantilever under weight loading are derived. The effects of including shear and longitudinal deformations are found to be negligible for configurations in which the two deformations tend to offset one another—such as at the onset of buckling under a compressive load—but are shown to be significant for certain configurations in which the two deformations are additive—as in some instances of weight loading.     

A new efficient method to evaluate exact stiffness and mass matrices of non-uniform beams resting on an elastic foundation

In this paper, a new efficient method to evaluate the exact stiffness and mass matrices of a non-uniform Bernoulli–Euler beam resting on an elastic Winkler foundation is presented. The non-uniformity may result from variable cross-section and/or from inhomogeneous linearly elastic material. It is assumed that there is no abrupt variation in the cross-section of the beam so that the Euler–Bernoulli theory is valid. The method is based on the integration of the exact shape functions which are derived from the solution of the axial deformation problem of a non-uniform bar and the bending problem of a non-uniform beam which are both formulated in terms of the two displacement components. The governing differential equations are uncoupled with variable coefficients and are solved within the framework of the analog equation concept. According to this, the two differential equations with variable coefficients are replaced by two linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under ideal load distributions. The key point of the method is the evaluation of the two ideal loads which in this work is achieved by approximating them by two polynomials. More specifically, the axial ideal load is approximated by a linear polynomial while the transverse one by a cubic polynomial. The numerical implementation of the method is simple, and the results are compared favorably to those obtained by exact solutions available in literature.     

中性线修正型变截面梁类构件压电控制

传统绝对节点坐标法(absolute nodal coordinate formulation,ANCF)在变截面梁类构件建模过程中常以几何中位线等效构造单元中性线,难以对变截面单元位移场状态进行精确描述.为解决此类问题,本文以中细型变截面梁类构件为研究对象,深入考虑变截面结构几何因素及复合材料属性对变截面梁类构件中性...     

Secondary Deformations Due to Axial Shearing of the Annular Region Between Two Eccentrically Placed Cylinders

Fosdick and Kao [1] extended a conjecture of Ericksen's [2] for non-linear fluids, to non-linear elastic solids, and showed that unless the material moduli of an isotropic elastic material satisfied certain special relations, axial shearing of cylinders would be necessarily accompanied by secondary deformations if the cross-section were not a circle or the annular region between two concentric circles. Further, they used the driving force as the small parameter for a perturbation analysis and showed that the secondary deformation will occur at fourth order, much in common with what is known for non-linear fluids. Here, we show that if on the other hand the driving force is not small (of O(1)), but the departure of the cylinder from circular symmetry is small, then secondary deformations appear at first order, the parameter for perturbance being the divergence from circular symmetry. This revised version was published online in August 2006 with corrections to the Cover Date.     

A Prestressed Elastic Strip with Elastic Reinforcements

A contact problem is studied for a prestressed elastic strip with an elastic reinforcement. The integral Fourier transform is used to construct an influence function for an infinite strip with one face fixed. A unit concentrated force is applied to the strip at an arbitrary angle. The contact problem on force transfer from a thin infinite stringer to the prestressed strip is solved. The problem is mathematically formulated as a system of integro-differential equations for the unknown contact stresses on the assumption that the beam bending model and the uniaxial stress model are valid for the stringer, which is subjected to both vertical and horizontal forces. This system is solved in a closed form using the integral Fourier transform. The contact stresses are expressed in terms of Fourier integrals in a quite simple form. The influence of the initial stresses on the contact stress distribution is analyzed, and effects of concentrated load are revealed     

On Λ-fractional elastic solid mechanics

After defining the fractional Λ-derivative, having all the requirements for corresponding to a differential, the fractional Λ-strain is established. Contrary to the common strain, that has a local character, fractional strain access a non-local character, quite important for expressing deformations in non-homogeneous media with microcracks and inhomogeneities, that may change during deformation. The purpose of the present work is the establishement of the principles and laws of the non-linear Λ-fractional Elasticity. The Λ-fractional non-linear stress–strain relations are derived. The restriction into the linear fields is presented. Further, fractional deformation of a fractal bar is discussed. The Fractional deformations and fractional elastic problems are set up with the definition of stresses and displacements in the initial space. Further, the Λ-fractional analysis with its conjugate Λ-fractional space is presented, considering fractional derivatives of both sides in the bending of a cantilever beam under uniform continuously distributed loading.

     

反倾层状岩体斜坡弯曲-拉裂两种失稳破坏之判据探讨

对反倾层状岩体斜坡弯曲-拉裂的失稳破坏判据,已有研究分别基于两种力学模型进行推导,即竖直压杆弹性屈曲稳定和平直梁弯折破坏模型,但对层间错动阻力及挠度产生附加弯矩等因素未加以考虑,不尽合理。在反倾斜坡岩层受力分析基础上,建立考虑了板侧层间错动阻力的下端嵌固、上端自由的斜置等厚弹性悬臂板梁模型,统一地通过瑞利-里兹能量方法,推导了弹性屈曲临界条件和嵌固端弯折破坏临界条件。实例计算及讨论表明,弹性屈曲判据适用于陡立岩层;而中-陡反倾岩层应主要为弯折破坏,但层间的力学性质对弯折临界判据值具有较大影响。     

Damage induced instability in beam bending

The effect of internal damage creation on the load carrying capacity of a beam is studied. Time independent relations are postulated for the development of strain and damage with increasing net stress. The resulting relations between load and deformation for a Bernoulli-Navier beam lamina with rectangular cross-section are then derived. A state of instability is shown to exist, characterized by unlimited rate of increase in deformation and damage with load. An instability locus in the plane of bending moment and normal force is defined. The shape of this locus is studied for varying parameters in the deformation and damage laws.     

Equilibrium of a Prestressed Strip Reinforced with Elastic Plates

The contact problem for a prestressed elastic strip reinforced with equally spaced elastic plates is considered. The Fourier integral transform is used to construct an influence function of a unit concentrated force acting on the infinite elastic strip with one edge constrained. The transmission of forces from the thin elastic plates to the prestressed strip is analyzed. On the assumption that the beam bending model and the uniaxial stress model are valid for an elastic plate subjected to both vertical and horizontal forces, the problem is mathematically formulated as a system of integro-differential equations for unknown contact stresses. This system is reduced to an infinite system of algebraic equations solved by the reduction method. The effect of the initial stresses on the distribution of contact forces in the strip under tension and compression is studied     

Large-strain constitutive laws for the cylindrical deformation of shells

Expressions are developed for the strain energy and stress resultants in the case of large strain-large bending cylindrical deformations of shells of non-linear materials. The deformation variables are the extension and curvature of the material reference surface. The Mooney material is used as an example where appropriate.The influence of the large deformations on the constitutive laws is evaluated. Of special interest is the effect of cross terms and of the ability to express the stress resultants as partial derivatives of the strain energy with respect to appropriate deformation variables.     

Non-linear closed-form computational model of cable trusses

In this paper the non-linear closed-form static computational model of the pre-stressed suspended biconvex and biconcave cable trusses with unmovable, movable, or elastic yielding supports subjected to vertical distributed load applied over the entire span and over a part (over the half) of the span is presented. The paper is an extension of the previously published work of authors [S. Kmet, Z. Kokorudova, Non-linear analytical solution for cable trusses, Journal of Engineering Mechanics ASCE 132 (1) (2006) 119-123]. Irvine's linearized forms of the deflection and the cable equations are modified because the effects of the non-linear truss behaviour needed to be incorporated in them. The concrete forms of the system of two non-linear cubic cable equations due to the load type are derived and presented. From a solution of a non-linear vertical equilibrium equation for a loaded cable truss, the additional vertical deflection is determined. The computational analytical model serves to determine the response, i.e. horizontal components of cable forces and deflection of the geometrically non-linear biconvex or biconcave cable truss to the applied loading, considering effects of elastic deformations, temperature changes and elastic supports. The application of the derived non-linear analytical model is illustrated by numerical examples. Resulting responses of the symmetric and asymmetric cable trusses with various geometries (shallow and deep profiles) obtained by the present non-linear closed-form solution are compared with those obtained by Irvine's linear solution and those by the non-linear finite element method. The conditions for the use of the linear and non-linear approach are briefly specified.     

Large deformations of in-plane beam

A theory of large deformations of straight slender in-plane beam is presented, based on the assumption of uniaxiality of the strain tensor. The integration of the compatibility conditions gives the parabolic variation of the axial component of the strain tensor over the cross-section of the beam. The integration of the kinematic equations gives linear variation of the displacement components over the cross-section. The equilibrium equations are written on a deformed configuration and numerically solved for linear elastic cantilever, subjected to concentrated or distributed conservative and nonconservative loads.     

开口厚壁截面短构件的约束扭转1）

为了分析开口厚壁截面短构件的约束扭转问题,采用统一分析梁模型与有限节线法,对T形和L形厚壁截面短构件约束扭转时横截面的翘曲和应力分布情况等问题进行了分析研究.算例计算结果表明：开口厚壁截面短构件存在与其横截面形心位置不一致的扭转（弯曲）中心,构件在不过扭转中心的外力作用下会产生弯扭耦合变形,其横截面将产生不均匀翘曲,横截面上的翘曲正应力和扭转剪应力均呈非线性分布.     

Numerical analysis of large elasto-plastic deflection of constant curvature beam under follower load

This paper describes a method to analyze the elasto-plastic large deflection of a curved beam subjected to a tip concentrated follower load. The load is made to act at an arbitrary inclination with the tip tangent. A moment-curvature based constitutive law is obtained from linearly hardening model. The deflection governing equation obtained is highly non-linear owing to both kinematics and material non-linearity. It is linearized to obtain the incremental differential equation. This in turn is solved using the classical Runge–Kutta 4^th order explicit solver, thereby avoiding iterations. Elastic results are validated with published literature and the new results pertaining to elasto-plastic cases are presented in suitable non-dimensional form. The load to end angle response of the structure is studied by varying normalized material and kinematic parameters. It is found that the response curves overlap at small deflection corresponding to elastic deformation and diverge for difference in plastic property. The divergent response curves intersect with each other at higher deflection. The results presented also show that the approach may be used to obtain desired non-uniformly curved beam by suitably loading a uniform curvature beam.