横观各向同性弹性半空间地基上正交异性矩形中厚板弯曲解析解

对横观各向同性体通解进行双重傅里叶变换,获得了直角坐标系下横观各向同性弹性半空间地基受任意竖向荷载作用下的位移积分变换解;在此基础上建立了板与地基的变形协调方程,并与三个广义位移变量描述的弹性地基上四边自由正交各向异性矩形中厚板的弯曲控制方程相结合,用三角级数法,得出横观各向同性弹性半空间地基上四边自由正交异性矩形中厚板受任意竖向荷载作用的弯曲解析解。相关算例分析表明,本文方法是有效的。     

A numerical method for clamped thin rectangular plates carrying a uniformly distributed load

A numerical method for clamped thin rectangular plates carrying a uniformly distributed load and the exact solution of the governing equations are given. The solution is presented in a simple form suitable for direct practical use. The method is a very simple and practical approach. The results are compared with those reported in the previous papers Published in Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 126–143, June 2007.     

四边固支矩形薄板自由振动的哈密顿解析解

在哈密顿体系中利用辛几何方法求解了四边固支矩形薄板自由振动问题的解析解。首先,从基本方程出发,将问题表示成Hamilton正则方程,然后利用辛几何方法导出本征值问题,从而得到本征函数解,使之满足边界条件;再由方程组有非零解的条件,最终推导出四边固支矩形薄板的自振频率方程,得到频率的解析解。计算了不同长宽比情况下四边固支矩形薄板的频率,结果与已有文献完全一致。该解法有望推广至更多尚未得到解析解的矩形板的振动问题。     

Benchmark trigonometric and 3-D elasticity solutions for an exponentially graded thick rectangular plate

The bending problem of a transverse load acting on an isotropic inhomogeneous rectangular plate using both two-dimensional (2-D) trigonometric and three-dimensional (3-D) elasticity solutions is considered. In the present 2-D solution, trigonometric terms are used for the displacements in addition to the initial terms of a power series through the thickness. The effects due to transverse shear and normal deformations are both included. The form of the assumed 2-D displacements is simplified by enforcing traction-free boundary conditions at the faces of the plate. No transverse shear correction factors are needed because a correct representation of the transverse shearing strain is given. The plate material is exponentially graded, meaning that Lamé’s coefficients vary exponentially in a given fixed direction (the thickness direction). A wide variety of results for the displacements and stresses of an exponentially graded rectangular plate are presented. The validity of the present 2-D trigonometric solution is demonstrated by comparison with the 3-D elasticity solution. The influence of aspect ratio, side-to-thickness ratio and the exponentially graded parameter on the bending response are investigated.     

Uniformly valid asymptotic solutions of the nonlinear unsymmetrical bending for orthotropic rectangular thin plate of four clamped edges with variable thickness

By using “the method of modified two-variable”,“the method of mixing perturbation ”and introducing four small parameters, the problem of the nonlinear unsymmetrical bending for orthotropic rectangular thin plate with linear variable thickness is studied. And the uniformly valid asymptotic solution of Nth- order for ε1 and Mth- order for ε2 of the deflection functions and stress function are obtained.     

The problems of nonlinear bending for orthotropic rectangular plate with four clamped edges

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Three dimensional elasticity solution for vibration problem of thick plate

I.Introduction'TheclassicalplatetheoryofvibrationisbasedontheKirchhoffhypotllcsis.andtheeffectoftransversesheardeformationandrotatoryinertiaareneglected.111[lltllcfreevibratiollfrequenciesofthinplates,allkindsofboundaryconditionshavebeensolved.Fortheanalysisofthevibrationalproblemsofthickplate,thetheoryofReissuer,MindlinandHenkyisused.Accordingtothedegreesoftheeffectsoftransverseshearstress,transversenormalstressandrotatoryinertia,somedynamicproblemsinEngineeringhavebeensolvedl:l.Regardless…     

Reciprocal theorem method for solving the problems of bending of thick rectangular plates

ＲＥＣＩＰＲＯＣＡＬＴＨＥＯＲＥＭＭＥＴＨＯＤＦＯＲＳＯＬＶＩＮＧＴＨＥＰＲＯＢＬＥＭＳＯＦＢＥＮＤＩＮＧＯＦＴＨＩＣＫＲＥＣＴＡＮＧＵＬＡＲＰＬＡＴＥＳＦｕＢａｏ－ｌｉａｎ（付宝连）Ｔａｎｗｅｎ－ｆｅｎｇ（谭文锋）（ＹａｎｓｈａｎＵｎｉｒｅｒｓｉｔ...     

四边自由各向异性矩形中厚板弯曲解析解

将基于阿穆巴诸米扬各向异性中厚板理论的控制方程、不同支撑下建立的板与支撑的协调方程以及板的自由边界条件,通过对称性分解成中心对称问题和中心反对称问题的叠加。利用傅立叶三角级数求解混合奇偶阶偏微分方程组,求得包括板的内力和支撑反力表达式的弯曲解析解。本文方法取消了直法线假设,克服了数学上的求解困难,去除了数值法的弊端,得出的结果更贴近实际。用该方法不但可以研究四边自由各向异性中厚板的弯曲特性和振动特性,而且还可分析不同边界约束下各向异性矩形中厚板的静动力特性。     

基于有限条带的厚/薄板矩形通用单元

基于两广义位移梁理论，利用解析试函数来建立厚／薄梁单元的横向位移、转角位移、曲率、剪应变等位移模式，构造出厚／薄梁通用单元．应用有限条带，将厚／薄梁单元的位移模式应用于厚／薄板矩形弯曲单元，直接构造出单元的横向位移、转角、曲率、剪应变，导出了单元的刚度矩阵和结点等效力，编制了计算程序，进行了数值计算和比较，结果表明，所研究的单元不出现剪切闭锁且精度较好．     

BENDING SOLUTION OF A RECTANGULAR PLATE WITH ONE EDGE BUILT－IN AND ONE CORNER POINT SUPPORTED SUBJECTED TO UNIFORM LOAD

ＢＥＮＤＩＮＧ　ＳＯＬＵＴＩＯＮ　ＯＦ　Ａ　ＲＥＣＴＡＮＧＵＬＡＲ　ＰＬＡＴＥ　ＷＩＴＨ　ＯＮＥ　ＥＤＧＥ　ＢＵＩＬＴ－ＩＮ　ＡＮＤ　ＯＮＥ　ＣＯＲＮＥＲ　ＰＯＩＮＴ　ＳＵＰＰＯＲＴＥＤ　ＳＵＢＪＥＣＴＥＤ　ＴＯ　ＵＮＩＦＯＲＭ　ＬＯＡＤＸｕＱｉｌｏ...     

层状横观各向同性地基上异性矩形薄板的弯曲解析解

选用更具广泛性的层状横观各向同性弹性地基模型,来分析四边自由各向异性矩形地基板的弯曲解析解。先基于直角坐标下横观各向同性体的静力胡海昌通解,借助双重傅里叶变换及矩阵传递法,获得层状横观各向同性地基的静力位移场和应力场;然后将异性薄板的弯曲控制方程,与基于层状横观各向同性弹性地基的位移解建立的板与地基变形协调方程相结合,先按对称性分解,再用三角级数法,得出层状横观各向同性弹性地基上四边自由各向异性矩形薄板的弯曲解析解,包括地基反力、板的挠度及板的内力的解析表达式。克服了数值法的弊端,取消了对地基反力的假设,且避免了矩阵指数函数的计算;同时考虑了地基的层状性及板和地基的各向异性,从而得到板的内力及地基反力更切实际的分布规律。算例结果与文献的有限元结果吻合良好,证明本文方法是切实可行的。     

Three-dimensional elasticity solutions for bending of generally supported thick functionally graded plates

Three-dimensional elasticity solutions for static bending of thick functionally graded plates are presented using a hybrid semi-analytical approach-the state-space based differential quadrature method （SSDQM）. The plate is generally supported at four edges for which the two-way differential quadrature method is used to solve the in-plane variations of the stress and displacement fields numerically. An approximate laminate model （ALM） is exploited to reduce the inhomogeneous plate into a multi-layered laminate, thus applying the state space method to solve analytically in the thickness direction. Both the convergence properties of SSDQM and ALM are examined. The SSDQM is validated by comparing the numerical results with the exact solutions reported in the literature. As an example, the Mori-Tanaka model is used to predict the effective bulk and shear moduli. Effects of gradient index and aspect ratios on the bending behavior of functionally graded thick plates are investigated.     

A solving method for a system of partial differential equations with an application to the bending problem of a thick plate

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Bending of uniformly loaded rectangular plates with two adjacent edges clamped, one edge simply supported and the other edge free

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一边固定一角点或二角点支承的矩形板弯曲统一求解方法

本文的统一法可以解决悬壁板或带有角点支承时在任意荷载作用下的弯曲。这种方法求解思路清晰、收敛速度快,计算精度高。     

铁磁矩形简支板的磁弹塑性弯曲行为分析

基于广义变分原理得到的磁力计算公式,采用塑性增量理论,Mises屈服准则和有效的增量有限元计算方法,研究了线性强化材料铁磁矩形板的磁弹塑性弯曲行为。在文中定量模拟了铁磁简支矩形板在外加磁场作用下的挠度特征曲线,铁磁板发生塑性变形时的构型图和不同外加磁场下的中截面构型,以及铁磁板在卸载后的残余挠度特征曲线等力学特征,分析了塑性区域随磁场增加而扩展的情况。数值结果表明:当铁磁矩形板上的部分区域发生塑性屈服后,其变形明显大于相同磁场条件下铁磁板发生的弹性变形值;且随着外加磁场倾角的增大(0°<α≤45°),铁磁板进入塑性屈服状态的临界屈服磁场值减小;铁磁板的中截面构形为双半波型,其塑性区域由铁磁板两侧挠度最大的区域向板的中心区域扩展,板的中心最后进入塑性区域等。     

An equation of motion for a thick viscoelastic plate

In this paper an equation of motion is presented for a general thick viscoelastic plate, including the effects of shear deformation, extrusion deformation and rotatory inertia. This equation is the generalization of equations of motion for the corresponding thick elastic plate, and it can be degenerated into several types of equations for various special cases.     

The derivation of exact static conditions at the corner points for the bending of thick rectangular plates

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