A new higher-order shear deformation theory and refined beam element of composite laminates

A new higher-order shear deformation theory based on global-local superposition technique is developed. The theory satisfies the free surface conditions and the geometric and stress continuity conditions at interfaces. The global displacement components are of the Reddy theory and local components are of the internal first to third-order terms in each layer. A two-node beam element based on this theory is proposed. The solutions are compared with 3D-elasticity solutions. Numerical results show that present beam element has higher computational efficiency and higher accuracy.

On well-posedness of two-phase nonlocal integral models for higher-order refined shear deformation beams

Due to the conflict between equilibrium and constitutive requirements,Eringen’s strain-driven nonlocal integral model is not applicable to nanostructures of engineering interest.As an alternative,the stress-driven model has been recently developed.In this paper,for higher-order shear deformation beams,the ill-posed issue(i.e.,excessive mandatory boundary conditions(BCs)cannot be met simultaneously)exists not only in strain-driven nonlocal models but also in stress-driven ones.The well-posedness of both the strain-and stress-driven two-phase nonlocal(TPN-Strain D and TPN-Stress D)models is pertinently evidenced by formulating the static bending of curved beams made of functionally graded(FG)materials.The two-phase nonlocal integral constitutive relation is equivalent to a differential law equipped with two restriction conditions.By using the generalized differential quadrature method(GDQM),the coupling governing equations are solved numerically.The results show that the two-phase models can predict consistent scale-effects under different supported and loading conditions.

Analysis of antisym metric cross-ply laminates using high-order shear deformation theories:a meshless approach

For many years finite element method(FEM)was the chosen numerical method for the analysis of composite structures.However,in the last 20 years,the scientific community has witnessed the birth and development of several meshless methods,which are more flexible and equally accurate numerical methods.The meshless method used in this work is the natural neighbour radial point interpolation method(NNRPIM).In order to discretize the problem domain,the NNRPIM only requires an unstructured nodal distribution.Then,using the Voronoi mathematical concept,it enforces the nodal connectivity and constructs the background integration mesh.The NNRPIM shape functions are constructed using the radial point interpolation technique.In this work,the displacement field of composite laminated plates is defined by high-order shear deformation theories.In the end,several antisymmetric cross-ply laminates were analysed and the NNRPIM solutions were compared with the literature.The obtained results show the efficiency and accuracy of the NNRPIM formulation.

Size-dependent sinusoidal beam model for dynamic instability of single-walled carbon nanotubes

In this study, a model for dynamic instability of embedded single-walled car- bon nanotubes （SWCNTs） is presented. SWCNTs are modeled by the sinusoidal shear deformation beam theory （SSDBT）. The modified couple stress theory （MCST） is con- sidered in order to capture the size effects. The surrounding elastic medium is described by a visco-Pasternak foundation model, which accounts for normal, transverse shear, and damping loads. The motion equations are derived based on Hamilton＇s principle. The differential quadrature method （DQM） in conjunction with the Bolotin method is used in order to calculate the dynamic instability region （DIR） of SWCNTs. The effects of differ- ent parameters, such as nonlocal parameter, visco-Pasternak foundation, mode numbers, and geometrical parameters, are shown on the dynamic instability of SWCNTs. The re- sults depict that increasing the nonlocal parameter shifts the DIR to right. The results presented in this paper would be helpful in design and manufacturing of nano-electromechanical system （NEMS） and micro-electro-mechanical system （MEMS）.

Thermo-mechanical Behaviors of Functionally Graded Shape Memory Alloy Timoshenko Composite Beams

This paper focuses on the thermo-mechanical behaviors of functionally graded(FG)shape memory alloy(SMA)composite beams based on Timoshenko beam theory.The volume fraction of SMA fiber is graded in the thickness of beam according to a power-law function and the equivalent parameters are formulated.The governing differential equations,which can be solved by direct integration,are established by employing the composite laminated plate theory.The influences of FG parameter,ambient temperature and SMA fiber laying angle on the thermo-mechanical behaviors are numerically simulated and discussed under different boundary conditions.Results indicate that the neutral plane does not coincide with the middle plane of the composite beam and the distribution of martensite is asymmetric along the thickness.Both the increments of the functionally graded parameter and ambient temperature make the composite beam become stiffer.However,the influence of the SMA fiber laying angle can be negligent.This work can provide the theoretical basis for the design and application of FG SMA structures.

General analytical solution to bending of composite laminated beams with delaminations

Based on the first-order shear deformable beam theory, a refined model for composite beams containing a through-the-width delamination is presented, and the deformation at the delamination front is considered. Different from the ordinary delami- nated beam theory, each of the perfectly bonded portions of the new model is constructed as two separated beams along the interface without assuming a plane section at the de- lamination front. The governing equations of the delaminated portions and bonded ones are established, combined with continuity conditions of displacements and internal forces. Solutions of delaminated composite beams with different boundary conditions, delamina- tion locations and sizes axe shown in excellent agreement with the finite element results, showing efficiency and applicability of the present model.

Determination of Shear Center of Arbitrary Cross-Section

In structural analysis, it is often necessary to determine the geometrical properties of cross section. The location of the shear center is greater importance for an arbitrary cross section. In this study, the problems of coupled shearing and torsional were analyzed by using the finite element method. Namely, the simultaneous equations with respect to the warping, shear deflection, angle of torsion and Lagrange’s multipliers are derived by finite element approximation. Solving them numerically, the matrix of the shearing rigidity and torsional rigidity is obtained. This matrix indicates the coupled shearing and torsional deflection. The shear center can be obtained determining the coordinate axes so as to eliminate the non-diagonal terms. Several numerical examples are performed and show that the present method gives excellent results for an arbitrary cross section.

基于n阶剪切变形理论的表面梯度脱碳汽车前轴弯曲分析

汽车前轴在热锻和热处理过程中,材料的表面会发生一定深度的脱碳,脱碳层的力学性能随脱碳深度呈梯度变化,影响了前轴在载荷下的弯曲性能。利用分段函数构建两侧表面功能梯度变化、内部性能均匀的简支夹芯梁,根据n阶剪切变形理论研究梁在两点载荷作用下的弯曲行为。利用虚功原理推导出位移场控制方程,采用Navier解析方法得到简支边界条件下梁的弯曲行为,并与相关文献中的实例进行比较。结果表明:n阶剪切变形理论具有良好的精度与可靠性;梁的挠度与转角随脱碳指数k的增大而增大,并于k≈10处达到准稳态;脱碳深度大于5 mm时,脱碳层厚度对梁弯曲时产生应力的影响比梁的高度变化带来的影响更大;两侧脱碳深度不对称时,物理中性面发生迁移;梁的弯曲挠度与转角随梁的厚度、宽度的增大而减小,但梁的厚度变化的影响更大。

双参数弹性基功能梯度圆柱管自由振动的高阶梁理论解

基于高阶剪切变形梁理论,推导了Winkler-Pasternak弹性基径向功能梯度空心圆柱管自由振动行为的控制方程。该方法无须引入剪切修正系数,自动满足空心圆柱管内外表面剪应力自由边界条件。通过引入辅助函数,将关于挠度和转角的耦合控制方程化为单一高阶微分控制方程。给出了典型边界条件下功能梯度空心圆柱管的频率和振型。将计算结果与已有文献结果对比,验证了所提理论的精度。可以为工程中常见的Winkler-Pasternak弹性基梁结构提供更高精度的一维弹性理论解,研究结果表明,功能梯度材料的梯度参数和弹性地基刚度系数对固有频率值影响显著。与高阶固有频率相比,刚度系数对低阶固有频率的影响更加明显。

高阶剪切变形理论下Pasternak地基上FG-CNTRC梁自由振动及屈曲分析

功能梯度碳纳米管增强复合材料(functionally graded carbon nanotube-reinforced composite,FG-CNTRC)作为新一代先进复合材料,因其优越的力学性能而被研究者们广泛关注。该研究以Pasternak地基上FG-CNTRC梁为研究对象,基于不同高阶剪切变形理论,将一种具有插值特性的无网格法——稳定移动克里金插值(stabilized moving Kriging interpolation,SMKI)应用于求解Pasternak地基上FG-CNTRC梁的自由振动及屈曲问题。基于稳定移动克里金插值和不同高阶剪切变形理论给出FG-CNTRC梁的位移场,利用Hamilton原理和最小势能原理分别推导了Pasternak地基上FG-CNTRC梁的自由振动和屈曲控制方程。采用MATLAB软件编制了相关程序,将该研究解与解析解或文献解对比,证明了该方法在计算Pasternak地基上FG-CNTRC梁自由振动与屈曲的有效性及准确性。最后,还讨论了不同高阶剪切变形理论、地基系数、碳纳米管体积分数等对其自振频率和屈曲临界荷载的影响。

Dynamic Loadings Induced Vibration of Third Order Shear Deformable FG-CNTRC Beams:Gram-Schmidt-Ritz Method

This research work deals with a study on dynamic behavior of functionally graded carbon nanotube-reinforced composite(FG-CNTRC)beams under various types of dynamic loads.Carbon nanotubes(CNTs)are used as the reinforcing materials that distribute continuously across the beam thickness.By using third order shear deformable theory(TSDT)in this current study,the straightness and normality of the transverse normal after deformation are unconstrained.The equations of motion based on TSDT are solved by Gram-Schmidt-Ritz method in which the displacement functions are generated via Gram-Schmidt procedure.Additionally,the time-integration of Newmark is also employed to carry out dynamic response of the beams under dynamic loads.Several effects such as material distributions,types of dynamic loads,boundary conditions and so on are taken into account.According to numerical results,it can be revealed that adding small amount of CNTs can reduce considerably the dynamic amplitude of FG-CNTRC beams.Moreover,the dynamic analysis of beam-like structures plays an important role in structural design because mass inertia matrix of the beam being involved in the equations of motion,which yields much larger deflection than that predicted by simple static analysis.

含分层复合材料层合梁弯曲问题的一般解法

基于一阶剪切梁理论,考虑分层边缘区域的变形特点,提出了含穿透分层复合材料梁模型.与传统分层模型不同,该文将未分层部分看作上下子梁,放弃了传统模型中分层前缘横截面始终保持平面的假设.通过分层前缘的位移连续条件和内力连续条件,建立了粘合段和分层段的控制方程.并且,应用该模型对不同边界条件下含不同分层尺寸对称和非对称分层的复合材料层合梁弯曲问题进行了求解,结果与三维有限元计算的结果一致,从而证明了模型的有效性和适用性.

基于一阶剪切变形理论的新型复合材料层合板单元

基于一阶剪切变形理论(FSDT),本文构造一种新型的20自由度(每结点5个自由度),四边形复合材料层合板单元,适合于任意铺设情形的层合板的计算。它是按如下方式构造的:(1) 单元每边的转角和剪应变由Timoshenko层合厚梁理论来确定;(2) 对单元域内的转角场和剪应变场进行合理的插值;(3) 引入平面内双线性位移场来体现层合板面内与弯曲的耦合作用。本文单元,记为TMQ20,不存在剪切闭锁现象,在计算单层的各向同性板时可以退化为文[1]中优质的中厚板单元TMQ。在文[2]中将给出本文单元对于层合板问题的详细数值算例。

考虑剪切变形效应的受竖向集中荷载作用的单跨斜梁研究

为解决中、小跨径斜梁结构分析中采用初等梁理论而不考虑剪切变形的影响而导致计算挠度偏小的问题,基于Timoshenko深梁理论的力法及图乘法,推导了考虑剪切变形影响的单跨斜深梁在竖向集中荷载作用下的内力和变形计算公式,并用有限元软件对所推导的计算公式进行了验证。结果表明:本文理论解析结果与有限元结果吻合较好,证明了本文理论公式的正确性。通过对标准A型单跨斜箱梁桥的跨径比、斜交角、平面形状等参数进行分析得到跨径越小、斜交角越大,剪切效应对斜梁挠度的影响越大;剪切效应对平行四边形斜梁的挠度影响最大,对等腰梯形斜梁的挠度影响最小;中、小跨径斜梁桥挠度计算时应采用考虑剪切变形影响的Timoshenko深梁理论,否则会低估挠度结果;所推导的斜深梁桥内力与变形计算公式能方便设计人员的使用,有助于推动斜梁桥计算理论的深化和拓展,丰富了斜梁桥的计算方法。

计及不同剪切变形的功能梯度材料梁的弯曲分析

研究矩形截面功能梯度材料(Functionally graded materials,FGM)梁在不同剪切变形理论下的静力弯曲问题。假设FGM梁由金属和陶瓷两种材料构成,其等效物性参数沿厚度方向连续变化,且遵从简单幂率变化规律。基于最小势能原理,建立以轴向位移、横向位移及转角为未知函数的FGM梁的运动微分方程组。对简支FGM梁,采用Fourier级数法获得5种剪切变形理论下FGM梁的挠度、轴向位移及转角曲线,分析梁的长高比、梯度指标对弯曲变形的影响,分析不同剪切变形理论下FGM梁的切应力和正应力的分布特性,并与均质材料梁的静力弯曲特性进行比较。给出FGM梁的中性轴位置随梯度指标的变化曲线并进行分析。

考虑剪切变形影响的桩基m法计算理论

考虑桩基的剪切变形影响,利用单广义位移深梁理论,建立了桩基m法的计算方法,导出了水平位移、转角、弯矩和剪力的初参数表达式和无量纲参数函数的统一表达式,根据桩底边界条件建立了初参数解的计算公式;给出了无量纲参数函数随换算深度和弯剪刚度比的变化图形。研究表明,换算深度小于3.0时,弯剪刚度比对无量纲参数函数影响较小,换算深度大于4.0时,弯剪刚度比对无量纲参数函数影响的趋势非常明显,桩基剪切变形的影响程度与桩的边界条件有关。算例结果表明,桩身的剪切变形有增大桩顶水平位移、提高弯矩零点位置、改变弯矩分布特征、扩大桩侧土压力大小等影响。

基于修正偶应力和高阶剪切变形理论的变截面微梁的自由振动

基于修正偶应力和高阶剪切理论建立了仅含有一个尺度参数的Reddy变截面微梁的自由振动模型,研究了变截面微梁自由振动问题的尺度效应和横向剪切变形对自振频率计算的影响。基于哈密顿原理推导了动力学方程与边界条件,并采用微分求积法求解了各种边界条件下的自振频率。算例结果表明,基于偶应力理论预测的变截面微梁的自振频率均大于经典梁理论的预测结果,即捕捉到了尺度效应。另外,梁的几何尺寸与尺度参数越接近,尺度效应就越明显,而梁的长细比越小,横向剪切变形对自振频率的影响就越明显。

用切比雪夫多项式研究短梁的弯曲计算

考虑剪切效应,利用切比雪夫多项式构造严格满足表面切应力边界条件的轴向位移表达式,建立了短梁弯曲问题的新理论.利用奇异函数把作用在短梁上的复杂外载荷表示为分布载荷,推导出了短梁弯曲时的截面正应力公式及挠曲线表达式.把采用切比雪夫多项式推导出短梁的弯曲计算公式计算结果与弹性理论计算结果进行比较,可知该方法的计算精度较高.研究结果表明:在复杂外载荷作用下,当长高比小于等于6时,剪切变形对梁的弯曲挠度影响较大,而当长高比小于3时,剪切变形对梁的弯曲应力影响较大;因此建议采用切比雪夫多项式方法给出的挠度表达式、弯曲应力进行计算,因为切比雪夫多项式方法不但给出了复杂外载荷作用下梁截面挠度、弯曲应力的计算通式,而且该方法具有计算过程简便、精度高的优点.

基于一阶剪切变形理论的变曲率曲梁的几何非线性方程

基于一阶剪切变形理论和轴线可伸长的精确几何非线性理论,推导了变曲率曲梁在热机载荷共同作用下的几何非线性控制方程。通过引入轴线伸长率,变形后的轴线弧长被当作基本未知量之一,基本未知量均被表示为变形前的轴线坐标的函数,使问题的求解区间仍为未变形时的曲梁轴线长度;给出了在给定曲梁轴线参数方程时,利用本文控制方程进行几何分析所需的初始曲率、变形前曲梁几何关系的数学表达式;介绍了几种常见的曲梁边界条件。所给数学模型可为轴线可伸长的变曲率曲梁的几何非线性分析和计算提供理论参考。

剪切可变形梁非线性静态响应的精确解

本文给出了纵横向载荷作用下,梁非线性静态问题的精确解。基于非线性一阶剪切变形梁理论,导出了梁非线性静态问题的基本方程。将三个非线性方程化简为一个关于横向挠度的非齐次四阶非线性积分-微分方程,当只有轴向载荷作用时,该方程和相应的边界条件构成微分特征值问题。直接求解该方程,得到了梁非线性静态变形闭合形式的解,这个解显式地给出了梁的变形与外载荷之间的非线性关系,描述了梁变形后的非线性平衡路径。利用这个解,得到了梁临界屈曲载荷的一阶结果与经典结果。为考察载荷、长高比以及边界条件的影响,根据得到的解析解给出了一些数值算例,并讨论了梁不同阶屈曲模态下非线性静态响应的一些性质。结果表明:对应于方程特征参数λ的不同取值区间,梁的轴向载荷-挠度曲线有不同的解支;而对应于参数λ的同一取值区间,梁分别对应两个不同的屈曲模态。