Bending of thick beams of bimodulus materials

The literature on bending of beams made of bimodulus materials (which have one value of the elastic modulus in tension and another in compression) is limited. All of the works known to the present investigators are restricted to beams with natural boundary conditions and subjected to specific distributions of normal pressure only. In the present work, the transfer-matrix approach is used to determine the small-deflection static behavior of bimodulus beams, including transverse shear deformation. The neutral surface, i.e. the locus of points having zero axial normal strain, may vary linearly within each element. The effects of axial load and non-natural boundary conditions are considered. As a basis for comparative evaluation, exact closed-form solutions are also presented for special cases in which the neutral-surface location is constant along the beam axis. Results are compared between the two solution methods and are found to give good agreement.     

基于欧拉 伯努利梁和铁木辛柯梁理论的功能梯度材料模量测定

为测定功能梯度材料的弹性模量和剪切模量,引入梁理论并将梁沿长度方向离散,建立单元平衡方程后可得到弹性模量和剪切模量分布;假设弹性模量为沿长度方向的线性函数或指数函数,用有限元软件仿真计算功能梯度材料梁单元节点处的挠度和转角,然后用插值法构造变形特征函数,并计算得出弹性模量和剪切模量,且计算值与理论值的误差较小.计算结果还表明,采用铁木辛柯梁理论不仅可以得到弹性模量,还可以计算剪切模量,且弹性模量计算结果比用欧拉-伯努利梁计算结果更接近真实值,但铁木辛柯梁理论中需测定转角,对测定过程的要求会更加严格。     

Influence lines for bending moments in beams on elastic foundations

A finite difference method assuming parabolic variation of contact pressure distribution is presented to obtain the influence lines for bending moments in beams on an elastic foundation. These influence lines can conveniently be used to find moments in beams on elastic foundations due to any type of loads. The computational procedure presented is simple. Accurate results are obtained with only 10 elements.     

Probabilistic computer analysis of circular rafts by the method of realizations

This paper deals with certain probabilistic aspects of statical behaviour of axisymmetric circular rafts resting on elastic media. In particular, the study concerns the problem of system randomness stemming from random variabilities of both the modulus of elasticity of the raft plate material, and the properties of the elastic media supporting the superstructure. Two types of homogeneous foundation bed are dealt with, namely, Winkler springs with a random subgrade modulus, and an elastic half-space whose Young's modulus is a random quantity. With the aid of the finite difference procedure in the variational formulation, the problems are solved by applying the so-called method of realizations. The Gaussian (normal) and a symmetrical beta distributions are considered in the work. The probabilistic analysis presented here involves relatively low computational costs and yields all the main characteristics of the discrete probabilistic fields of the system structural response.     

A finite element analysis of beams on elastic foundation including shear and axial effects

A displacement finite element method for analyzing a beam on continuous elastic foundation is presented. A three-dimensional model which accounts for the effects of both the Filonenko-Borodich and Pasternak foundation models in a consistent and complete way is used. A variational principle is introduced with the slope field due to bending only and the displacement field approximated by independent quantities subjected to variation. Numerical examples illustrate the accuracy of the element, the importance of shear, axial and shear-axial interaction effects associated with continuous elastic foundation, and finally the application of the element to a rotor supported by two hydrodynamic journal bearings.     

Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model

This paper aims to investigate the size scale effect on the buckling and post-buckling of single-walled carbon nanotube (SWCNT) rested on nonlinear elastic foundations using energy-equivalent model (EEM). CNTs are modelled as a beam with higher order shear deformation to consider a shear effect and eliminate the shear correction factor, which appeared in Timoshenko and missed in Euler–Bernoulli beam theories. Energy-equivalent model is proposed to bridge the chemical energy between atoms with mechanical strain energy of beam structure. Therefore, Young’s and shear moduli and Poisson’s ratio for zigzag (n, 0), and armchair (n, n) carbon nanotubes (CNTs) are presented as functions of orientation and force constants. Conservation energy principle is exploited to derive governing equations of motion in terms of primary displacement variable. The differential–integral quadrature method (DIQM) is exploited to discretize the problem in spatial domain and transformed the integro-differential equilibrium equations to algebraic equations. The static problem is solved for critical buckling loads and the post-buckling deformation as a function of applied axial load, CNT length, orientations and elastic foundation parameters. Numerical results show that effects of chirality angle, boundary conditions, tube length and elastic foundation constants on buckling and post-buckling behaviors of armchair and zigzag CNTs are significant. This model is helpful especially in mechanical design of NEMS manufactured from CNTs.

     

Elastic instability of a beam resting on an elastic foundation subjected to a partially tangential force

The influence of the tangency coefficient, the elastic foundation and the elastically restrained boundary conditions on the elastic instability of a uniform Bernoulli-Euler beam is investigated. It is shown that at both ends of the beam, if any one of the two elastic spring constants is infinite then the tangency coefficient will have no influence on the critical load of the beam. For a clamped-elastic springs supported beam, it is found that the regions of flutter instability increase, when the elastic foundation modulus is increased and the tangential coefficient may either increase or reduce the stability of the system, depending upon the elastic foundation modulus, the translational spring and the rotational spring constants.     

Analysis of beams on non-homogeneous elastic foundation

An approximate method is developed to study the bending of the beam on non-homogeneous elastic foundation. The solutions are obtained by translating the differential equations into the integral equations and applying the numerical integrations. It is shown that we can analyze the beams with arbitrary boundary conditions and load conditions resting on non-homogeneous elastic foundation.     

A nonlinear solution scheme for multistory, multibay plane frames

If a plane frame experiences bending deformations in the fundamental state or is in a post-buckled secondary state, then a precise determination of the load-deflection relationships requires consideration of member rotations. This leads to highly nonlinear equations: and for multistory, multibay frames with many members, the solution is difficult to obtain. This paper presents a systematic procedure for handling the geometrically nonlinear equations, leading to an exact engineers' beam theory solution for rectangular plane frames with linearly elastic material. A procedure for locating the critical load is also discussed.     

Exact dynamic stiffness matrix of non-symmetric thin-walled beams on elastic foundation using power series method

Exact dynamic element stiffness matrix for the flexural–torsional free vibration analysis of the shear deformable thin-walled beam with non-symmetric cross-section on two-types of elastic foundation is newly presented using power series method based on the technical computing program Mathematica. For this, the shear deformable beam on elastic foundation theory is developed by introducing Vlasov's assumption and applying Hellinger–Reissner principle. This beam includes the shear deformation effects due to the shear forces and the restrained warping torsion and due to the coupled effects between them, and rotary inertia effects and the flexural–torsional coupling effects due to the non-symmetric cross-sections. And then equations of motion and force–deformation relations are derived from the energy principle and explicit expressions for displacement parameters are derived based on power series expansions of displacement components and the exact dynamic element stiffness matrix is determined using force–deformation relationships. In order to verify the accuracy of this study, the numerical solutions are presented and compared with the analytical solutions and the finite element solutions using the isoparametric beam elements. Particularly the influences of the coupled shear deformation on the vibrational behavior of non-symmetric beam on elastic foundation are investigated.     

薄片介质高速传动的动态仿真与分析

针对薄片介质在金融自动处理设备中传动和堆叠过程中常见的卡阻故障,采用RecurDyn计算薄片介质的力学特性,获得薄片介质的弹性模量与弯曲挠度之间的关系曲线,以用于确定不同新旧程度薄片介质的弹性模量;基于票券自动处理设备中的堆叠模块,对介质在传动过程中的变形和运动姿态进行仿真,计算支撑固件的不同长度所对应的介质弯曲挠度.结果表明：支撑固件长度越大,介质产生的弯曲挠度越大,有利于避免介质堆叠时发生卡阻问题.     

A twin shear beam model for double chord RHS joints

A finite element elasto-plastic analysis of a twin shear beam model, developed to shed light on the behavior of separated double chord gap joints of rectangular hollow sections (RHS), is presented. The model treats a single beam member as composed of a plate simulating its inner web and a channel representing top and bottom flanges and the outer web. The twelve degrees of freedom (DOF) nonconforming plate bending element and the eight DOF plane stress elements are used to form a plate element for the inner web, while the channel consists of a grillage of beam elements. The analysis, extended beyond the elastic limit, accounts for strain hardening and material nonlinearity. A comparison of theoretical predictions with the experimental results of 24 specimens shows generally good agreement. The mean ratios of theoretical to experimental values are 0.97 and 0.87 for elastic stiffness and carrying capacity measured at a post elastic deformation limit.     

Stress distributions of short fiber composite materials

In the present study, the normal and interfacial shear stress distributions of short fiber composites under a force parallel to the fiber direction are investigated by using the global-local finite element method. A finite element analytical procedure will be developed by combining the hybrid stress finite element method and the conventional displacement-based plane stress element to predict the normal and interfacial shear stress of short fiber composites. The effects of different geometrical shapes of fiber end, the ratio of fiber length to diameter, the fiber volume fraction and the elastic modulus ratio to the normal stress and the interfacial shear stress distributions of short fiber composites will also be conducted. The geometrical shapes of fiber end under present investigation are blunt, V-shaped, wedge-shaped and semi-circular, respectively.     

Multi-objective optimization of functionally graded materials,thickness and aspect ratio in micro-beams embedded in an elastic medium

Optimal design of micron-scale beams as a general case is an important problem for development of micro-electromechanical devices. For various applications, the mechanical parameters such as mass, maximum deflection and stress, natural frequency and buckling load are considered in strategies of micro-manufacturing technologies. However, all parameters are not of equal importance in each operating condition but multi-objective optimization is able to select optimal states of micro-beams which have desirable performances in various micro-electromechanical devices. This paper provides optimal states of design variables including thickness, distribution parameter of functionally graded materials, and aspect ratio in simply supported FG micro-beams resting on the elastic foundation using analytical solutions. The elastic medium is assumed to be as a two-layered foundation including a shear layer and a linear normal layer. Also, the size effect on the mechanical parameters is considered using the modified strain gradient theory and non-dominated sorting genetic algorithm-II is employed to optimization procedure. The target functions are defined such that the maximum deflection, maximum stress and mass must be minimized while natural frequency and critical buckling load must be maximized. The optimum patterns of FG micro-beams are presented for exponential and power-law FGMs and the effect of theory type and elastic foundation discussed in details. Findings indicate that the elastic foundation coefficients and internal length scale parameters of materials have the significant influences on the distribution of design variables. It is seen that the optimum values of inhomogeneity parameter and aspect ratio for E-FG micro-beams predicted by the modified strain gradient theory are larger than those of the classical continuum theory. Also, the multi-objective optimization is able to improve the normalized values of mass, maximum deflection, buckling load and natural frequency of P-FG micro-beams.     

Optimization of geometry described by curves

In CAD, the shape of a design is often described by an interpolating curve. In this paper, an automated method for finding the optimal geometry of such a curve is used to study the blade geometry of vertical axis wind turbines. To minimize bending moments due to centrifugal loads during rotation the blades are often curved in the shape of the troposkien (Greek for ‘turning rope’). From a fabrication point of view a simpler shape is preferred. An optimal number of interpolating points along the curve are used as design variables and their movements controlled by an optimization algorithm. During the optimization, the shape of the curve is successively recomputed and current objective and constraint functions are analysed. Stresses along the beam are computed using the finite element method by regarding the curve as a plane elastic beam.     

The analysis of nonlinear three dimensional frames

The problems involved in the analysis of three dimensional frames of non-prismatic members, made of nonlinearly elastic material are investigated. The sources of nonlinearity considered are material behavior and interaction of loads and small deformations. The material response is expressed in the form of a table of uniaxial stress and related strain with monotonically increasing values. This table is used to create a table of bending moment-curvature relationships for beams at a discrete number of cross-sections along their length and also a table of the axial load-biaxial moments-biaxial curvatures relationships for eccentrically loaded columns. These tables are then used to obtain the responses of structural members to different loadings. Length effects in the columns and sidesway in the frame are both considered. The Displacement Method of structural analysis is used with an incremental loading approach to analyze three dimensional frames made of materials whose stress-strain curves were assumed to be of circular or parabolic form, at different load patterns and magnitudes. It was found that the combination of three dimensional and nonlinear material effects on the magnitudes of the frame moments is important when compared to the moments obtained by linear, plane frame analysis, and that the assumption of linear behavior may give results which are in error by more than 30 per cent for high axial loads on columns.

It appears that the method, developed herein, of representing material and member behavior in tabular form, offers a very direct and versatile approach to the problem of analyzing structures made of several different and/or nonlinearly elastic materials.     

Finite element program for solution of geotechnical problems

A program, named RODSIM, based on the finite element displacement method has been developed, tested and applied to the solution of major geotechnical problems. It is specially suitable to analyze deformations due to sequential excavation of soil supported by diaphragm walls with anchors or struts in which case it produces the bending moments, shear forces, axial forces, horizontal and vertical pressures along the wall, after deformation. All element matrices are evaluated by exact integration considering the Young's modulus of the orthotropic, cross-anisotropic or isotropic material varying linearly within the six node triangular element. The node numbering system may be optimized automatically.     

Neural networks in mechanics of structures and materials – new results and prospects of applications

Basic ideas of back-propagation neural networks (BPNNs) are presented in short. Then BPNN applications in analysis of the following problems are discussed: (1) bending analysis of elastoplastic beams, (2) elastoplastic plane stress problem, (3) estimation of fundamental vibration periods of real buildings, (4) detection of damage in a steel beam, (5) identification of loads applied to an elastoplastic beam. Regularization neural network is briefly discussed and its application to estimation of concrete fatigue durability it shown. A modified Hopfield network is used to the analysis of an elastic angular plate with unilateral constraints. In the end some conclusions and prospects of neurocomputing applications are pointed out.     

Elastic analysis of frames considering panel zones deformations

Structural analysis of frames is generally based on centreline-to-centreline geometry. However in many structures, member dimensions might be quite large and have a significant effect on frame lateral stiffness. Rigid offsets extending from the joint centrelines to the faces of the members are used to model the rigid rotation kinematic of finite joint area. This approach can underestimate drifts of structures since it neglects elastic deformations of the joint regions. In this paper, a joint element is used to consider the rigid kinematic motion, elastic shear, and bending deformations of beam-column panel zone regions. Parametric analyses have been carried out on the seismic response of a ten-storey steel frame to determine the impact of panel zone deformations on the structure's elastic response. It is shown that kinematic effects and panel zone shear deformations are best modelled using rigid zone reduction factors to define rigid offset effective lengths. On the other hand, the consideration of panel zone bending deformations resulted in drifts values that were significantly larger than the usual rigid point joint assumption.