Analysis of spatial beamlike lattices with rigid joints

Micropolar beam models are developed for the static, free vibration and buckling analysis of repetitive spatial beamlike lattices with rigid joints. The micropolar beam models have independent microrotation and displacement fields and are characterized by their strain energy, potential energy due to initial stresses and kinetic energy from which the governing differential equations and boundary conditions can be derived. The procedure for developing the expression for the strain energy of the micropolar beam involves introducing basic assumptions regarding the variation of the displacement and microrotation components in the plane of the cross-section, and obtaining effective elastic coefficients of the continuum in terms of the material properties and geometry of the original lattice structure. The high accuracy of the solutions obtained by the micropolar beam models is demonstrated by means of numerical examples for vierendeel and double-laced lattice girders with triangular cross-sections.     

硅纳米梁振动基波频率的半连续体模型

基于Sun-Zhang提出的应变能量计算模型,根据动势能转换与守恒原理,提出了用于分析硅纳米梁基波频率的半连续体模型。与传统的连续体模型相比,该方法考虑了厚度方向进入纳米尺度所带来的物理特性的离散化现象。本文对理论推得的基频模型进行了验证,在极小尺寸时,使用分子动力学软件MaterialStudioTM验证,在较大尺寸时,用连续介质模型验证,从验证结果可以看出,在这两种尺寸下,各自都很吻合,说明该模型可以适用于从纳米尺寸到宏观尺寸的所有尺度范围。     

Analysis of beam-like lattice trusses

A simple procedure is presented for predicting the thermoelastic and free vibration responses of large repetitive beam-like trusses. The procedure is based on replacing the original lattice structure by an equivalent continuum beam model and obtaining closed-form (exact) solutions for the beam model. The equivalent beam model accounts for warping and shear deformation in the plane of the cross-section and is characterized by its thermoelastic strain and kinetic energies, from which the equations of motion and constitutive relations can be derived. The high accuracy of the results obtained by the proposed approach is demonstrated by means of numerical examples.     

硅纳米梁的静态特性分析

提出了用于分析硅纳米梁静态弯曲的半连续体模型.与传统的连续体模型相比,该方法考虑了厚度方向进入纳米尺度所带来的物理特性的离散化现象.基于Sun-Zhang提出的应变能量计算模型,运用变分原理,推导出半连续体模型.从计算结果可以看出,几何尺寸和表面原子的重构、弛豫效应对梁的弯曲有一定影响.该模型分析梁的弯曲得到的结果与连续体模型相比偏小,随着尺寸的增大误差逐渐减小,在宏观尺寸下两种模型最终趋于一致.     

An efficient size-dependent shear deformable shell model and molecular dynamics simulation for axial instability analysis of silicon nanoshells

Understanding the size-dependent behavior of structures at nanoscale is essential in order to have an effective design of nanosystems. In the current investigation, the surface elasticity theory is extended to study the nonlinear buckling and postbuckling response of axially loaded silicon cylindrical naoshells. Thereby, an efficient size-dependent shear deformable shell model is developed including the size effect of surface free energy. A boundary layer theory of shell buckling in conjunction with a perturbation-based solution methodology is employed to predict the size dependency in the buckling loads and postbuckling behavior of silicon nanoshells having various thicknesses. After that, on the basis of the Tersoff empirical potential, a molecular dynamics (MD) simulation is performed for a silicon cylindrical nanoshell with thickness of four times of silicon lattice constant, the critical buckling load and critical shortening of which are extracted and compared with those of the developed non-classical shell model. It is demonstrated that by taking the effects of surface free energy into account, a very good agreement is achieved between the results of the developed size-dependent continuum shell model and those of MD simulation.     

Micropolar beam models for lattice grids with rigid joints

A simple, rational approach is presented for developing micropolar beam models for large repetitive beam-like planar lattices with rigid joints. The micropolar beam models have independent microrotation, and displacement fields and are characterized by their strain and kinetic energies, from which the equations of motion and boundary conditions can be derived. The procedure for developing the expression for the strain energy of the micropolar beam involves introducing basic assumptions regarding the variation of the displacement and microrotation components in the plane of the cross-section and obtaining effective elastic coefficients of the continuum in terms of the material properties and geometry of the original lattice structure. The high accuracy of the solutions obtained by the micropolar beam models is demonstrated by means of numerical examples.     

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.

     

Geometry optimization of a slender cantilever beam subjected to lateral buckling

The paper deals with the geometry optimization of a slender cantilever beam subjected to a concentrated force acting at the free end. The two-parametric mathematical model of lateral torsional buckling is based on the Bernoulli–Euler beam theory and is given in dimensionless form. The optimization procedure is performed using the optimal control theory and the relation between state and adjoint variables is presented. The boundary value problem derived from the optimization procedure is solved numerically and compared to solutions obtained via an alternative optimization approach called sequential approximate optimization.     

轴向冲击下波纹腹板梁的失稳控制和能量吸收

针对Sine波纹腹板梁结构,通过设计其初始振幅缺陷,控制不同波形阶数下波纹梁的失稳模态,引导上下翼板按预期渐进、稳定、可重复的压溃变形模式发展.采用Abaqus/Explicit中的显式动力学方法数值模拟初始振幅缺陷与比吸能(Specific Energy Absorption,SEA)的关系.通过对比低阶、中阶和高阶波形的载荷-位移曲线说明波形阶数对吸能特性和失稳控制的影响.结果表明:对于给定的波形阶数,存在初始振幅缺陷的临界值使Sine波纹腹板梁结构达到预期的临界失稳状态,同时SEA最大;初始振幅缺陷越大,失稳控制越容易,但吸能效果随之降低.     

Lateral buckling of locally buckled beams using finite element techniques

This paper deals with lateral-torsional buckling of beams which have already buckled locally before the occurrence of overall buckling. Due to the weakening effects of local buckling, the stiffness of the beam is reduced. As a result, overall lateral buckling takes place at a lower load than the member would carry in the absence of local buckling. The effective width concept is used in this investigation to account for the post-buckling strength in the buckled compression plate elements of the beam section. A finite element formulation in conjunction with effective width concept is presented. Due to the nonlinearity involved because of local buckling, an iterative procedure is necessary. Search techniques are used to find the load factor. The method combined with an analysis on nonlinear bending moment distribution can be used to analyze the lateral stability problem of locally buckled continuous structure. In this case, both elastic stiffness matrix and geometric stiffness matrix must be revised at each load level. A computer program has been prepared for an IBM 370/165 computer.     

Optimum energy-based design of BRB frames using nonlinear response history analysis

In this paper, an optimum design method for buckling restrained brace frames subjected to seismic loading is presented. The multi-objective charged system search is developed to optimize costs and damages caused by the earthquake for steel frames. Minimum structural weight and minimum seismic energy which including seismic input energy divided by maximum hysteretic energy of fuse members are selected as two objective functions to find a Pareto solutions that copes with considered preferences. Also, main design constraints containing allowable amount of the inter-story drift and plastic rotation of beam, column members and plastic displacement of buckling restrained braces are controlled. The results of optimum design for three different frames are obtained and investigated by the developed method.     

Benchmark case studies in optimization of geometrically nonlinear structures

A structural optimization algorithm is developed for truss and beam structures undergoing large deflections against instability. The method combines the nonlinear buckling analysis using the displacement control technique, with the optimality criteria approaches. Several benchmark case studies illustrate the procedure and the results are compared with examples reported in the literature. It is shown that a design based on the generalized eigenvalue problem (linear buckling) highly underestimates the optimum mass or overestimates the buckling load for these types of structures, so a design based on the linear buckling analysis may result in catastrophic failure. The effect of geometrical nonlinearities and element imperfections has also been studied.     

Spatial stability of shear deformable curved beams with non-symmetric thin-walled sections. II: F. E. solutions and parametric study

In the companion paper, an improved formulation for spatial stability analysis of shear deformable thin-walled curved beams with non-symmetric cross-sections is presented based on the displacement field considering both constant curvature effects and the second-order terms of semi-tangential rotations. Thus the elastic strain energy and the potential energy due to initial stress resultants are consistently derived. Also closed-form solutions for in-plane and lateral-torsional buckling of curved beams subjected to uniform compression and pure bending are newly derived for mono-symmetric thin-walled curved beams under simply supported and clamped end conditions. In this paper, F. E. procedures are developed by using curved and straight beam elements with non-symmetric cross-sections. Analytical and numerical solutions for spatial buckling of shear deformable thin-walled circular beams are presented and compared in order to illustrate the accuracy and the practical usefulness of this study. In addition, the extensive parametric studies are performed on spatial stability behavior of curved beams. Particularly transition and crossover phenomena of buckling mode shapes with change in curvature and length of beam on buckling for curved beams are investigated for the first time.     

Finite element model of ionic nanowires with size-dependent mechanical properties determined by ab initio calculations

A finite element procedure for modeling crystalline nanostructures such as nanowires is proposed. The size effects exhibited by nanoobjects are captured by taking into account a surface energy, following the classical Gurtin–Murdoch surface elasticity theory. An appropriate variational form and a finite element approach are provided to model and solve relevant problems numerically. We describe a simplified technique based on projection operators for constructing the surface elements. The methodology is completed with a computational procedure based on ab initio calculations to extract elastic coefficients of general anisotropic surfaces. The FEM continuum model is validated by comparisons with complete ab initio models of nanowires with different diameters where size-dependent mechanical properties are observed. The FEM continuum model can then be used to model similar nanostructures in ranges of sizes or geometries where analytical or atomistic model is limited. The validated model is applied to the analysis of size effects in the bending of an AlN nanowire.     

On the role of geometrically exact and second-order theories in buckling and post-buckling analysis of three-dimensional beam structures

Several geometrically nonlinear beam models are evaluated with respect to their utility in the analysis of buckling and post-buckling behavior of three-dimensional beam structures. The first two models are based on the so-called geometrically exact beam theory capable of representing finite rotations and finite displacements. The principal difference between these models concerns only the chosen parameterization of finite rotations, with the orthogonal matrix used in the first and the rotation vector used in the second one. The third beam model based on the second-order approximation of finite rotations is also discussed along with its application to constructing a consistent formulation of the linear eigenvalue problem for computing an estimate of the critical load. Exact linearized forms, which are crucial for facilitating the buckling load computation and assuring a robust performance of a Newton-method-based continuation strategy, are presented for all three beam models. An elaborate set of numerical simulations of buckling and post-buckling analysis of beam structures is given in order to illustrate the performance of each of the presented models. Finally, some conclusions are drawn.     

Nonlocal strain gradient finite element analysis of nanobeams using two-variable trigonometric shear deformation theory

In the present paper, a new trigonometric two-variable shear deformation beam nonlocal strain gradient theory is developed and applied to investigate the combined effects of nonlocal stress and strain gradient on the bending, buckling and free vibration analysis of nanobeams. The model introduces a nonlocal stress field parameter and a length scale parameter to capture the size effect. The governing equations derived are solved employing finite element method using a 3-nodes beam element, developed for this purpose. The predictive capability of the proposed model is shown through illustrative examples for bending, buckling and free vibration of nanobeams. Comparisons with other higher-order shear deformation beam theory are also performed to validate its numerical implementation and assess its accuracy within the nonlocal context.

     

Computation of lower-bound elastic buckling loads using general-purpose finite element codes

This paper investigates ways to have a computational implementation of a lower bound approach for the buckling of imperfection-sensitive shells using general purpose finite element codes. This approach was developed by Croll and others, and has been mainly employed by developing special purpose programs or analytical solutions. However, it is felt that this limits the possibilities of the user, and this shortcoming is addressed in the paper. First, the formulation is presented in a way to highlight what computations can be done following a reduced energy approach. Then, a methodology is implemented in conjunction with a general purpose program to compute the lower bound buckling load for cylindrical shells with different geometric configurations under uniform pressure. The accuracy of the procedure and the difficulties in the implementation, depending on the finite element chosen for the discretization are shown. Results demonstrate that the proposed reduced energy model can predict the lower bound load for cylindrical shells under uniform pressure distributions.     

Dynamic analysis of geometrically nonlinear truss structures

Beam-like truss structures undergoing large deflections when subjected to static and dynamic loadings are studied by using the matrix method and equivalent continuum models. For the matrix method, an incremental procedure with equilibrium iterations are used. Equivalent continuum beam models are derived based on the properties of a typical substructure of the truss. Solutions obtained by using both methods are compared for a number of examples.     

Finite element analysis of out-of-plane buckling of reinforced concrete walls

In this paper a FE model for the study of the out-of-plane buckling of reinforced concrete walls is derived. The concrete is modelled using non-linear orthotropic 16 d.f. plate bending elements; the reinforcing steel using elasto-plastic beam elements. In the plane of the structure stresses are calculated using either four or eight-node membrane elements with bar elements used for the reinforcing steel. The buckling load is calculated by determining when the determinant of the out-of-plane tangent stiffness becomes zero. Comparison of the FE model with available experimental results shows good agreement.     

Self-buckling of micromachined beams under resistive heating

Self-buckling behavior of micromachined beams under resistive heating is described by an electromechanical model with experimental verifications. This model consists of both electrothermal and thermoelastic analyses for beam-shape polysilicon microstructures that are fabricated by a standard surface micromachining process. When an input electrical current is applied, Joule-heating effects trigger the thermal expansion of beam structures and cause mechanical buckling. The standard testing devices are clamped-clamped bridges, 2-μm wide, 2-μm thick, and 100-μm long. It is found that a minimum current of 3.5 mA is required to cause beam buckling. Under an input current of 4.8 mA, a lateral deflection of 2.9±0.2 μm at the center of the bridge is measured with a computer image processing scheme. The experimental measurements are found to be consistent with analytical predictions. A discussion of modeling considerations and process variations is presented