关于一阶剪切板的K-rm-n型精化理论

依据Mindlin-Reissner理论,着重研究一阶剪切板的K-rm-n型精化理论,并推导出仅以挠度和应力函数为未知变量的广义K-rm-n型大挠度方程,适用于复合材料上复合构造剪切板的非线性分析.在当前板的精化理论中,消去的两个转角以隐含形式作用于板的整体变形.这一工程理论适用于各种计及横向剪切复合材料板、正交各向异性中厚板和夹层板等的线性和非线性分析.容易发现,针对具体的工程应用,由该方程可直接获得相应的退化形式,并且与文献所载的一致.     

关于一阶剪切板的Karman型精化理论

依据Ｍｉｎｄｌｉｎ－Ｒｅｉｓｓｎｅｒ理论,着重研究一阶剪切板的Ｋａｒｍａｎ型推导出仅以挠度和应力函数为未知变量的广义Ｋａｒｍａｎ型大挠度方程,适用于复合材料上复合构造的切板的非线性分析,在当前板的数理化理论中,消支的两个转角以隐含形式作用于板的整体变形,这一工程理论适用于各种计及横向剪切复合材料板、正交各向异性中厚板和层板等的线性和非线性分析。容易发现,针对具体的工程应用,由该方程可直接获得相应的     

关于一阶剪切板的Kármán型精化理论

依据Mindlin_Reissner理论 ,着重研究一阶剪切板的K rm n型精化理论 ,并推导出仅以挠度和应力函数为未知变量的广义K rm n型大挠度方程 ,适用于复合材料上复合构造剪切板的非线性分析·　在当前板的精化理论中 ,消去的两个转角以隐含形式作用于板的整体变形·　这一工程理论适用于各种计及横向剪切复合材料板、正交各向异性中厚板和夹层板等的线性和非线性分析·　容易发现 ,针对具体的工程应用 ,由该方程可直接获得相应的退化形式 ,并且与文献所载的一致·     

中厚板的弹性屈曲和后屈曲

本文采用Reissner假定考虑横向剪切变形的影响,导出弹性矩形板大挠度方程.本文讨论考虑横向剪切变形的矩形板的弹性屈曲和后屈曲.采用文[8]提供的摄动方法,给出了完善和非完善中厚板的后屈曲平衡路径,并与经典薄板理论结果进行了比较.     

基于曲率插值的大变形梁单元

线性梁单元的形函数在单元大转动时会引起虚假应变,不适用于几何非线性分析．传统的几何非线性梁单元由于位移插值和转角插值的相干性,常常引起剪切闭锁等问题．该文 提出了一种平面大变形梁单元,通过单元域内的曲率插值以及曲率与节点位移之间的函数关系,将单元节点力和节点位移表示为节点曲率的函数．由于曲率插值本质上是对梁的应变进行插值,保证了单元任意刚体运动不会产生虚假的节点力;且将梁的截面形心位移表示为曲率的函数,避免了传统单元中的剪切闭锁问题．因而所提方法特别适用于梁的几何非线性分析．数值算例说明了所提方法的正确性和有效性.     

单箱双室简支箱梁剪切变形及剪力滞双重效应分析

基于各个翼板选取不同的最大剪切转角差为剪力滞广义位移,应用能量变分原理分别推导出了考虑和不考虑剪切变形时单箱双室截面控制微分方程组,结合边界条件给出了箱梁纵向应力和竖向挠度的初参数解,从力学、数学角度上证实了剪切变形和剪力滞效应是两个相对独立的力学行为,进一步阐述了二者对箱梁的影响,即剪切变形对箱梁截面纵向应力无影响,但是对竖向挠度有很大的影响.数值算例表明,利用该文解和数值解分析跨中截面剪力滞系数横向分布规律,二者吻合程度良好,其横向分布规律与单室箱梁类似,唯独不同之处是边腹板处的剪力滞效应比中腹板处的剪力滞效应略微大一些;挠度计算表明,剪切效应使得该箱梁在集中和均布荷载作用下跨中挠度分别增大4.6%和2.7%.     

基于4变量精确平板理论的剪切效应分析

由于中厚板壳结构或者复合材料板壳结构中横向剪切模量与面内伸缩模量之比较低,剪切效应对其结构力学行为的影响较为严重.该文基于4变量精确平板理论(refined plate theory,RPT),对平板弯曲过程中的剪切效应进行了分析和讨论.首先对RPT进行了简要的介绍,然后以受横向均布载荷的四边简支矩形板为例,进行了无量纲的数值计算,着重讨论了平板的几何尺寸和性质对其剪切效应的影响:剪切效应随着厚度增加而加强,尤其是当宽厚比小于10以后,剪切效应会迅速加强;剪切效应随横向剪切模量与面内伸缩模量的比值减小而增强;在同种条件下,"剪切"挠度在正交各向异性板中的比重始终大于其在各向同性板中的比重,并且平板越厚或者长宽比越大,两种板的剪切效应差异性越大;各向同性板中的剪切效应受长宽比的变化表现得更加敏感,其"剪切"挠度比重随长宽比的增大而持续减小,而正交各向异性板中的"剪切"挠度比重则呈现出先减小后增大的趋势.研究结果可为提高板壳结构弯曲分析的准确性提供参考.     

正交各向异性体梁弯曲的弹性理论

本文由文献[1]横观各向同性板的弯曲弹性理论关于二维问题的特例,通过比拟,得到了正交各向异性梁弯曲的弹性理论,文中给出了求解正交各向异性梁弯曲问题的一种方法.提出了一种新的深梁理论,并指出了考虑横向剪切变形影响的Reissner理论对于应力分量的近似程度较差.     

基于修正偶应力理论的Timoshenko微梁模型和尺寸效应研究

基于修正偶应力理论,将Timoshenko微梁的应力、偶应力、应变、曲率等基本变量,描述为位移分量偏导数的表达式.根据最小势能原理,推导了决定Timoshenko微梁位移场的位移场控微分方程.利用级数法求解了任意载荷作用下Timoshenko简支微梁的位移场控微分方程,得到了反映尺寸效应的挠度、转角及应力的偶应力理论解.通过对承受余弦分布载荷Timoshenko简支微梁的数值计算,研究了Timoshenko微梁的挠度、转角和应力的尺寸效应,分析了Poisson比对Timoshenko微梁力学行为及其尺寸效应的影响.结果表明:当截面高度与材料特征长度的比值小于5时,Timoshenko微梁的刚度和强度均随着截面高度的减小而显著提高,表现出明显的尺寸效应;当截面高度与材料特征长度的比值大于10时,Timoshenko微梁的刚度与强度均趋于稳定,尺寸效应可以忽略;材料Poisson比是影响Timoshenko微梁力学行为及尺寸效应的重要因素,Poisson比越大Timoshenko微梁刚度和强度的尺寸效应越显著.该文建立的Timoshenko微梁模型,能有效描述Timoshenko微梁的力学行为及尺寸效应,可为微电子机械系统（MEMS）中的微结构设计与分析提供理论基础和技术参考.     

新型空间薄壁梁单元

基于Timoshenko梁理论和Vlasov薄壁杆件约束扭转理论,建立了具有内部结点的新型空间薄壁截面梁单元．通过对弯曲转角和翘曲角采取独立插值的方法,考虑了横向剪切变形,扭转剪切变形及其耦合作用,弯曲变形和扭转变形的耦合以及二次剪应力等因素影响,由Hellinger-Reissner广义变分原理,推得单元刚度矩阵．算例表明所建模型具有良好的精度,可用于空间薄壁杆系结构的有限元分析．     

求解位移方程的能量原理

本文在Castigliano原理的基础上推广了单位虚载荷法.据此,用力法直接导出了梁、板和壳一类结构的挠曲面的一般方程.我们导出了具有非齐次位移边界条件的矩形薄板和考虑横向切变形影响的矩形厚板的挠曲面方程.同时给出了相应直梁的挠曲轴方程.推广了互等定理的应用.计算了三个简例.     

Influence of the Lateral Eigenstrains on the Transverse Displacement of Wide Beams

The present paper studies the influence of lateral eigenstrains on the transverse deflection of wide beams. We show that in this case a laterally nonuniform transverse displacement becomes notable; moreover, it turns out that the axial variation of the transverse displacement is significantly altered in comparison to the results obtained from beam theory. In order to derive a corrected analytical solution for the transverse displacement of wide beams, the latter are modeled as thin plates with induced eigenstrains in both in–plane directions. A Galerkin method is utilized to solve the plate equations, in which solutions for the transverse displacement resulting from beam theory are used as shape functions for the plate deflection. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

非均匀Reissner板弯曲的精确元法

本文在阶梯折算法和精确解析法的基础上,提出构造有限元的新方法——精确元法.该方法不用变分原理,可适用于任意变系数正定和非正定偏微分方程.利用该方法,得到Reissuer板弯曲的一个非协调单元,它具有十五个自由度.由于节点位移参数仅含有挠度和转角,因此处理任意边界条件非常容易.文中给出证明,位移和内力均收敛于精确解.由精确元法所得到的单元不仅能用于厚板,也可用于薄板.文末给出四个算例.算例表明,利用本文的方法,可获得满意的结果,并有较高的数值精度.     

Static analysis of functionally graded beams using higher order shear deformation theory

Displacement field based on higher order shear deformation theory is implemented to study the static behavior of functionally graded metal–ceramic (FGM) beams under ambient temperature. FGM beams with variation of volume fraction of metal or ceramic based on power law exponent are considered. Using the principle of stationary potential energy, the finite element form of static equilibrium equation for FGM beam is presented. Two stiffness matrices are thus derived so that one among them will reflect the influence of rotation of the normal and the other shear rotation. Numerical results on the transverse deflection, axial and shear stresses in a moderately thick FGM beam under uniform distributed load for clamped–clamped and simply supported boundary conditions are discussed in depth. The effect of power law exponent for various combination of metal–ceramic FGM beam on the deflection and stresses are also commented. The studies reveal that, depending on whether the loading is on the ceramic rich face or metal rich face of the beam, the static deflection and the static stresses in the beam do not remain the same.     

Higher-order theories for symmetric and unsymmetric fiber reinforced composite beams with C finite elements

A simple C⁰ isoparametric finite element formulation based on a set of higher-order displacement models for the analysis of symmetric and asymmetric multilayered composite and sandwich beams subjected to sinusoidal loading is presented. These theories do not require the usual shear correction coefficients which are generally associated with the Timoshenko theory. The four-noded Lagrangian cubic element with kinematic models having four, five and six degrees of freedom per node is used. A computer algorithm is developed which incorporates realistic prediction of transverse interlaminar stresses from equilibrium equations. By comparing the results obtained with the elasticity solution and the CPT (classical laminated plate theory) it is shown that the present higher-order theories give a much better approximation to the behaviour of laminated composite beams, both thick and thin. In addition numerical results for unsymmetric sandwich beams are presented which may serve as benchmark for future investigations.     

Mathematical solution for bending of short hybrid composite beams with variable fibers spacing

In this study, the static response is presented for a simply supported functionally graded hybrid beam subjected to a transverse uniform load. Material properties of the beam are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. By varying the fiber volume fraction within a symmetric laminated beam and combining two fiber types to create a hybrid functionally graded material (FGM) can offer desirable increases in axial and bending stiffness. The equations governing the hybrid FGM beams are determined using the principle of virtual work (PVW) arising from the higher order shear deformation theories. Numerical results on the transverse deflection, axial and shear stresses in a moderately thick hybrid FGM beam under uniform distributed load are discussed in depth. The effect of power-law exponent on the deflection and stresses are also commented.     

Mindlin板条中弹性波传播问题的分析

基于Mindlin弹性厚板理论，采用Hamilton状态空间求解方法，研究了板条横向两侧为自由边界条件下的弹性：波传播问题，给出了问题的分析解，以及各种振动模态和波传播模的存在性．采用本征函数展开法给出了条形板中导波模的频散方程，并与直接根据Mindlin提出的解获得的频散关系进行了比较，通过比较发现：在短波时，频散曲线相差较大，即前几阶低频模式有一定差别。根据Hamilton体系得到的截止频率高一些，长波时频散曲线几乎相同，即高频模式无差别。     

Asymptotically exact joining conditions at the junction of plates with very different characteristics

The phenomenon of the boundary layer which occurs when plates are joined is studied A procedure for deriving the asymptotically exact joining (transmission) conditions which associate the two-dimensional equations for the deformation of the plates along the joining line Γ is developed using the method of matched asymptotic expansions. Two situations are discussed in which these conditions turn out to be non-standard: the bending moment in Γ must disappear and the deflection can undergo a jump (for real values of the physical parameters, the longitudinal displacements and forces as well as the bending and the shearing force always remain continuous). One of the situations (the joining of “thick, soft” and a “thin, rigid” shells) is characteristic of a moving loudspeaker system. The results of a numerical experiment, which confirm the asymptotic analysis of the problem, are presented.     

Analytical solutions of refined plate theory for bending,buckling and vibration analyses of thick plates

Analytical solutions for bending, buckling, and vibration analyses of thick rectangular plates with various boundary conditions are presented using two variable refined plate theory. The theory accounts for parabolic variation of transverse shear stress through the thickness of the plate without using shear correction factor. In addition, it contains only two unknowns and has strong similarities with the classical plate theory in many aspects such as equations of motion, boundary conditions, and stress resultant expressions. Equations of motion are derived from Hamilton’s principle. Closed-form solutions of deflection, buckling load, and natural frequency are obtained for rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions. Comparison studies are presented to verify the validity of present solutions. It is found that the deflection, stress, buckling load, and natural frequency obtained by the present theory match well with those obtained by the first-order and third-order shear deformation theories.     

The equivalence of the refined theory and the decomposition theorem of rectangular beams

This paper presents the decomposition theorem of rectangular beams and indicates that the general state of stress of beams can be decomposed into two parts: the interior state and the Papkovich–Fadle state (shortened form the P–F state). The refined theory of beams is derived by using Papkovich–Neuber solution (shortened form the P–N solution) and Lur’e method without ad hoc assumptions. It is shown that the displacements and stresses of the beam can be represented by the angle of rotation and the deflection of the neutral surface. Based on the refined beam theory, the exact equations for the beam without transverse surface loadings are derived and consist of two governing differential equations: the fourth-order equation and the transcendental equation. It is then proved that the refined beam theory and the decomposition beam theorem are equivalent, i.e., the fourth-order equation and the transcendental equation are equivalent to the interior state and the P–F state, respectively.