Non‐linear spatial Timoshenko beam element with curvature interpolation

The paper presents a spatial Timoshenko beam element with a total Lagrangian formulation. The element is based on curvature interpolation that is independent of the rigid‐body motion of the beam element and simplifies the formulation. The section response is derived from plane section kinematics. A two‐node beam element with constant curvature is relatively simple to formulate and exhibits excellent numerical convergence. The formulation is extended to N‐node elements with polynomial curvature interpolation. Models with moderate discretization yield results of sufficient accuracy with a small number of iterations at each load step. Generalized second‐order stress resultants are identified and the section response takes into account non‐linear material behaviour. Green–Lagrange strains are expressed in terms of section curvature and shear distortion, whose first and second variations are functions of node displacements and rotations. A symmetric tangent stiffness matrix is derived by consistent linearization and an iterative acceleration method is used to improve numerical convergence for hyperelastic materials. The comparison of analytical results with numerical simulations in the literature demonstrates the consistency, accuracy and superior numerical performance of the proposed element. Copyright © 2001 John Wiley & Sons, Ltd.     

含裂纹悬臂梁的振动与疲劳耦合分析

基于Paris方程和同步分析方法考虑振动与疲劳裂纹扩展耦合之影响,提出一种含裂纹梁的振动疲劳寿命分析思路.振动分析过程中,利用线性弹簧等效裂纹段,复弹性模量引入阻尼损耗因子,得到考虑裂纹扩展、激励频率和阻尼等因素影响的动应力响应.结果表明:裂纹扩展、激励频率和阻尼等因素对疲劳寿命具有重要的影响.通过振动分析与疲劳裂纹扩展寿命估算同步进行,可进一步提高疲劳寿命估算精度.     

A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation

Based on a modified couple stress theory, a model for composite laminated beam with first order shear deformation is developed. The characteristics of the theory are the use of rotation–displacement as dependent variable and the use of only one constant to describe the material’s micro-structural characteristics. The present model of beam can be viewed as a simplified couple stress theory in engineering mechanics. An example as a cross-ply simply supported beam subjected to cylindrical bending loads of f_w = q₀ sin (πx/L) is adopted and explicit expression of analysis solution is obtained. Numerical results show that the present beam model can capture the scale effects of microstructure, and the deflections and stresses of the present model of couple stress beam are smaller than that by the classical beam mode. Additionally, the present model can be reduced to the classical composite laminated Timoshenko beam model, Isotropic Timoshenko beam model of couple stress theory, classical isotropic Timoshenko beam, composite laminated Bernoulli–Euler beam model of couple stress theory and isotropic Bernoulli–Euler beam of couple stress theory.     

Timoshenko梁能量守恒逐步积分算法

该文提出了Timoshenko梁非线性动力分析的能量守恒逐步积分算法。采用共旋技术考虑结构的几何非线性,空间离散采用相关插值形式,避免了剪切锁定现象。在时间离散时利用多参数修正方法对等效的节点动力平衡方程进行修正,实现了离散系统在保守荷载作用下的能量守恒。算法具备二阶局部精度,与已有的平均加速度方法和隐式中点方法相比,具有更好的数值稳定性。在二维情形下与Simo方法对比,指出了Simo方法在受保守外弯矩作用时系统能量不守恒。最后,通过三个数值模拟算例验证了算法的性能和能量守恒特性。     

A nonlocal higher-order curved beam finite model including thickness stretching effect for bending analysis of curved nanobeams

In the present work, a finite element approach is developed for the static analysis of curved nanobeams using nonlocal elasticity beam theory based on Eringen formulation coupled with a higher-order shear deformation accounting for through-thickness stretching. The formulation is general in the sense that it can be used to compare the influence of different structural theories, through static and dynamic analyses of curved nanobeams. The governing equations derived here are solved introducing a 3-nodes beam element. The formulation is validated considering problems for which solutions are available. A comparative study is done here by different theories obtained through the formulation. The effects of various structural parameters such as thickness ratio, beam length, rise of the curved beam, loadings, boundary conditions, and nonlocal scale parameter are brought out on the static bending behaviors of curved nanobeams.     

An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nanobeams using a nonlocal elasticity model

This paper is concerned with the free transverse vibration of cracked nanobeams modeled after Eringen's nonlocal elasticity theory and Timoshenko beam theory. The cracked beam is modeled as two segments connected by a rotational spring located at the cracked section. This model promotes discontinuities in rotational displacement due to bending which is proportional to bending moment transmitted by the cracked section. The governing equations of cracked nanobeams with two symmetric and asymmetric boundary conditions are derived; then these equations are solved analytically based on concerning basic standard trigonometric and hyperbolic functions. Besides, the frequency parameters and the vibration modes of cracked nanobeams for variant crack positions, crack ratio, and small scale effect parameters are calculated. The vibration solutions obtained provide a better representation of the vibration behavior of short, stubby, micro/nanobeams where the effects of small scale, transverse shear deformation and rotary inertia are significant.     

Coupling effects in bending, buckling and free vibration of generally laminated composite beams

A closed form expression to determine the effective flexural modulus of a laminated composite beam is developed and presented in this contribution. This effective flexural modulus is applied to the bending, buckling and free vibration response of generally laminated composite beams with various boundary supports. The expression was developed using the combination of the Euler–Bernoulli beam and classical lamination theory. In addition the results of an extensive finite element analysis are used to validate the analytical model. The comparison of the analytical results, the finite element results and the experimental results showed good correlation. It is also observed that coupling response is an important variable that must be included in the computation of the effective flexural stiffness of generally laminated beam.     

A new analysis of nonlinear free vibration behavior of bi-functionally graded sandwich plates using the p-version of the finite element method

Geometrically nonlinear vibration of bi-functionally graded material (FGM) sandwich plates has been carried out by the p-version of the finite element method (FEM). The bi-FGM sandwich plate is made up of two face-sheet layers of two different FGM and one layer of homogeneous core. The nonlinear equations of motion of bi-FGM sandwich plates are establish using the harmonic balance method and solved iteratively by the linearized updated mode method. The effects of amplitude vibration, mechanical properties, geometrical parameters, thickness ratio of bi-FGM layers, and volume fraction exponent on the nonlinear vibration behavior of bi-FGM sandwich plates are plotted and investigated.     

Energy expressions and free vibration analysis of a rotating double tapered Timoshenko beam featuring bending–torsion coupling

In this study, free vibration analysis of a rotating, double tapered Timoshenko beam featuring coupling between flapwise bending and torsional vibrations is performed. At the beginning of the study, kinetic and potential energy expressions of a rotating Timoshenko beam having single cross-sectional symmetry are derived in a detailed way by using several explanatory tables and figures. In the following section, Hamilton’s principle is applied to the derived energy expressions to obtain the governing differential equations of motion. The parameters for the hub radius, rotational speed, rotary inertia, shear deformation, slenderness ratio, bending–torsion coupling and taper ratio are incorporated into the equations of motion. In the solution part, an efficient mathematical technique, called the differential transform method (DTM), is used to solve the governing differential equations of motion. Using the computer package, Mathematica, the mode shapes are plotted, the effects of the incorporated parameters on the natural frequencies are investigated. The calculated results are tabulated in several tables and plotted in several graphics.     

S型悬臂梁柔度性能的计算与分析

基于线弹性和小变形假设理论,以卡氏第二定理为理论基础,推导了S型悬臂梁x向、y向和z向柔度的解析计算公式,利用有限元方法对柔度解析式进行校验。通过定义柔度比函数,比较了倒角S型悬臂梁、直梁S型悬臂梁和圆弧S型悬臂梁的柔度性能。结果表明:S型悬臂梁各柔度计算公式的相对误差均在10%以内,理论分析与仿真结果基本吻合,验证了S型悬臂梁各柔度解析式的正确性。当倒角S型悬臂的γ=0.2时,圆弧S型悬臂梁的x向和z向柔度最大,直梁S型悬臂梁的y向柔度最大。当α=18,β=6时,直梁S型悬臂梁的y向柔度最大,圆弧S型悬臂梁的z向柔度最大;倒角S型悬臂的参数0.2≤γ≤2.2时,圆弧S型悬臂梁的x向柔度最大;倒角S型悬臂的参数γ>2.2时,倒角S型悬臂梁的x向柔度最大。本文的研究内容为S型悬臂梁的工程设计和应用提供了理论基础。     

Numerical convergence of nonlinear nonlocal continuum models to local elastodynamics

We quantify the numerical error and modeling error associated with replacing a nonlinear nonlocal bond‐based peridynamic model with a local elasticity model or a linearized peridynamic model away from the fracture set. The nonlocal model treated here is characterized by a double‐well potential and is a smooth version of the peridynamic model introduced in the work of Silling. The nonlinear peridynamic evolutions are shown to converge to the solution of linear elastodynamics at a rate linear with respect to the length scale ε of nonlocal interaction. This rate also holds for the convergence of solutions of the linearized peridynamic model to the solution of the local elastodynamic model. For local linear Lagrange interpolation, the consistency error for the numerical approximation is found to depend on the ratio between mesh size h and ε. More generally, for local Lagrange interpolation of order p≥1, the consistency error is of order h^p/ε. A new stability theory for the time discretization is provided and an explicit generalization of the CFL condition on the time step and its relation to mesh size h is given. Numerical simulations are provided illustrating the consistency error associated with the convergence of nonlinear and linearized peridynamics to linear elastodynamics.     

作大范围运动的柔性梁的动力学分析

对附着在空间运动体上的柔性悬臂梁的动力学进行了研究，利用微元法建立了中心刚体作任意三维大位移运动时柔性悬臂梁作横向和纵向振动的动力学方程，此动力学方程计及了动力刚化效应。在对柔性梁离散求解时考虑了横向弯曲对纵向变形的影响，最后通过几个例子分析了运动基上柔性梁的动力学行为。     

A unified nonlocal formulation for bending,buckling and free vibration analysis of nanobeams

Abstract

A unified nonlocal formulation is developed for the bending, buckling, and vibration analysis of nanobeams. Theoretical formulations of eighteen nonlocal beam theories are presented by using unified formulation. Small scale effect is considered based on the nonlocal differential constitutive relations of Eringen. The governing equations of motion and associated boundary conditions of the nanobeam are derived using Hamilton's principle. Closed form solutions are presented for a simply supported boundary condition using Navier's solution technique. Numerical results for axial and transverse shear stress are first time presented in this study which will serve as a benchmark for the future research.     

Non-linear vibration analysis of a functionally graded Timoshenko beam under action of a moving harmonic load

In this paper, non-linear dynamic analysis of a functionally graded (FG) beam with pinned–pinned supports due to a moving harmonic load has been performed by using Timoshenko beam theory with the von-Kármán’s non-linear strain–displacement relationships. Material properties of the beam vary continuously in thickness direction according to a power-law form. The system of equations of motion is derived by using Lagrange’s equations. Trial functions denoting transverse, axial deflections and rotation of the cross-sections of the beam are expressed in polynomial forms. The constraint conditions of supports are taken into account by using Lagrange multipliers. The obtained non-linear equations of motion are solved with aid of Newmark-β method in conjunction with the direct iteration method. In this study, the effects of large deflection, material distribution, velocity of the moving load and excitation frequency on the beam displacements, bending moments and stresses have been examined in detail. Convergence and comparison studies are performed. Results indicate that the above-mentioned effects play a very important role on the dynamic responses of the beam, and it is believed that new results are presented for non-linear dynamics of FG beams under moving loads which are of interest to the scientific and engineering community in the area of FGM structures.     

Numerical simulation of rotating bending process for U-tubes in heat exchangers

Heat exchangers comprise thousands of tubes having U-shaped portions. Rotating bending method has been widely utilized to make U-bends. Although this method shows an excellent performance, cracks have been frequently detected in the U-bends due to residual stresses induced by bending. In this paper, the bending process is simulated based on elastic–plastic finite element analyses in order to investigate the magnitude and distribution of the residual stresses including the effects of operating pressure. Analyses results show that the residual stress increases as the radius of U-bend decreases and that operating pressure has a detrimental effect in terms of stress corrosion cracking at the intrados of U-bend. It is thought that these results can be utilized for the estimations of fracture mechanics parameters such as limit load, stress intensity factor and J-integral, prevention of the cracking, and establishment of the optimum inspection strategy for the heat exchanger tubes.     

Free vibration analysis of thick laminated plates using a mixed finite element

Abstract

A mixed finite element scheme based on assumed local high‐order displacements is proposed for the free vibration of thick laminated plates. The effects of transverse shear deformation, transverse normal stress and rotary inertia are considered in the formulation. Cross‐ply laminates with simple supports and angle‐ply laminates with clamped edges are presented as examples. The three dimensional elasticity solutions of cross‐ply laminates with simple supports are used to assess the accuracy of the present scheme. The effects of the span‐to‐thickness, aspect and material anisotropy ratio on the fundamental natural frequency are investigated. The present results are compared with the results in the published literature, and agree closely with the 3‐D elasticity solutions.     

Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates

In this paper, nonlinear static and free vibration analysis of functionally graded piezoelectric plates has been carried out using finite element method under different sets of mechanical and electrical loadings. The plate with functionally graded piezoelectric material (FGPM) is assumed to be graded through the thickness by a simple power law distribution in terms of the volume fractions of the constituents. Only the geometrical nonlinearity has been taken into account and electric potential is assumed to be quadratic across the FGPM plate thickness. The governing equations are obtained using potential energy and Hamilton’s principle that includes elastic and piezoelectric effects. The finite element model is derived based on constitutive equation of piezoelectric material accounting for coupling between elasticity and electric effect using higher order plate elements. The present finite element is modeled with displacement components and electric potential as nodal degrees of freedom. Results are presented for two constituent FGPM plate under different mechanical boundary conditions. Numerical results for PZT-4/PZT-5H plate are given in dimensionless graphical forms. Effects of material composition and boundary conditions on nonlinear response are also studied. The numerical results obtained by the present model are in good agreement with the available solutions reported in the literature.     

On the eigenvalues of a uniform cantilever beam carrying any number of spring–damper–mass systems

The eigenvalues of a uniform cantilever beam carrying any number of spring–damper–mass systems with arbitrary magnitudes and locations were determined by means of the analytical‐and‐numerical‐combined method (ANCM). First of all, each spring–damper–mass system was replaced by a massless effective spring with spring constant k_eff, which is the main point that the ANCM is available for the present problem. Next, the equation of motion for the ‘constrained’ beam (with spring–damper–mass systems attached) was derived by using the natural frequencies and normal mode shapes of the ‘unconstrained’ beam (without carrying any attachments) incorporated with the expansion theorem. Finally, the equation of motion for the ‘constrained’ beam in ‘complex form’ is separated into the real and the imaginary parts. From either part, a set of simultaneous equations were obtained. Since the simultaneous equations are in ‘real form’, the eigenvalues of the ‘constrained’ beam were determined with the conventional numerical methods. To confirm the reliability of the presented theory, all the numerical results obtained from the ANCM were compared with the corresponding ones obtained from the conventional finite element method (FEM) and good agreement was achieved. Because the order of the property matrices for the equation of motion derived by using the ANCM is much lower than that by using the conventional FEM, the storing memory and the CPU time required by the ANCM are much less than those required by the FEM. Besides, the solution of the equation of motion derived from the ANCM can always be obtained with the general personal computers, but that from the FEM can sometimes be obtained only with the computers of workstations or main frames when the total degrees of freedom exceeding a certain limit. Copyright © 1999 John Wiley & Sons, Ltd.     

Finite strip buckling and free vibration analysis of stepped rectangular composite laminated plates

A general spline finite strip method is presented which allows the spline knots to be located arbitrarily along the plate strip and also facilitates the use of analytical integration in evaluating strip properties. The development takes place in the contexts of first‐order shear deformation plate theory and of classical plate theory, and encompasses composite laminated material. The prediction of natural frequencies and buckling stresses of stepped rectangular plates is considered using the new approach in which refinement of knot spacings is used local to a step change. The superstrip concept is used as part of an efficient solution procedure. A number of applications demonstrate the validity and practicability of the developed method. Copyright © John Wiley & Sons, Ltd.     

Scattering of waves by flaws in anisotropic laminated plates

The strip element method (SEM) is used to investigate wave scattering by rectangular flaws in anisotropic laminated plates. The plates containing flaws are divided by junctions into domains in which the SEM is applied. For each domain, SEM equations are obtained which give a relationship between the traction and displacement vector on the vertical boundaries. A set of equations which gives a relationship between the traction and displacement vector on the junctions is then obtained by assembling the SEM equations for all the domains. This set of equations is solved by using the conditions on the junctions. Scattered wave fields in the frequency domain for isotropic and composite plates with rectangular flaws are computed and discussed in comparison with results for corresponding plates without flaws. A technique for determining the length of a rectangular flaw in a plate is also presented. The results presented in this paper are of importance and could be used in the characterization of flaws in anisotropic laminated plates.