初始纵横向荷载效应对薄板自振频率的影响分析

首先基于能量变分原理,给出了同时考虑初始纵向、横向荷载效应情况下板的应变能表达、动力平衡微分方程及单元刚度矩阵。再以动力平衡微分方程为基础,运用伽辽金法解得四种典型板同时考虑初始纵向、横向荷载效应的基频近似解,并运用瑞利法解得简支矩形板考虑初始纵向、横向荷载效应的前三阶频率近似解。然后相互验证了考虑初始纵向、横向荷载效应的板单元刚度矩阵和频率近似解的正确性,并进一步分析了初始纵向、横向荷载及相关因素对板自振频率的影响。结果表明,考虑初始纵向、横向荷载效应后,板的自振频率主要受初始纵向荷载、初始横向荷载、板的厚度及边界条件等因素的影响;初始纵向、横向荷载效应对板的基频影响明显于高阶频率;初始纵向、横向荷载对板的自振频率影响分别呈线性和抛物线规律。     

初始纵横向荷载效应对薄板自振频率的影响分析The analysis for the effect of both initial in-plane and transverse loads on the natural frequency of plates

首先基于能量变分原理,给出了同时考虑初始纵向、横向荷载效应情况下板的应变能表达、动力平衡微分方程及单元刚度矩阵。再以动力平衡微分方程为基础,运用伽辽金法解得四种典型板同时考虑初始纵向、横向荷载效应的基频近似解,并运用瑞利法解得简支矩形板考虑初始纵向、横向荷载效应的前三阶频率近似解。然后相互验证了考虑初始纵向、横向荷载效应的板单元刚度矩阵和频率近似解的正确性,并进一步分析了初始纵向、横向荷载及相关因素对板自振频率的影响。结果表明,考虑初始纵向、横向荷载效应后,板的自振频率主要受初始纵向荷载、初始横向荷载、板的厚度及边界条件等因素的影响;初始纵向、横向荷载效应对板的基频影响明显于高阶频率;初始纵向、横向荷载对板的自振频率影响分别呈线性和抛物线规律。     

功能梯度矩形板的三维弹性分析

将功能梯度三维矩形板的位移变量按双三角级数展开，以弹性力学的平衡方程为基础．导出位移形式的平衡方程。引入状态空间方法，以三个位移分量及位移分量的一阶导数为状态变量，建立状态方程。考虑四边简支的边界条件，由状态方程得到了功能梯度三维矩形板的静力弯曲问题和自由振动问题的精确解。由给出的均匀矩形板自由振动问题的计算结果表明．与已有的理论解以及有限元方法的计算结果相吻合。假设功能梯度三维矩形板的材料常数沿板的厚度方向按照指数函数的规律变化．进一步给出了功能梯度三维矩形板的自由振动问题和静力弯曲问题的算例分析，并讨论了材料性质的梯度变化对板的动力响应和静力响应的影响。     

集中静荷载初始效应对固支梁固有频率的影响

研究了集中静荷载初始效应对于固支梁固有频率的影响,利用能量守恒方程和变分原理,得到了存在初始挠度时梁的振动控制方程。在此基础上,分析得到了存在初始挠度时梁固有频率的解析解,以及存在初始集中静荷载时固支梁的固有频率和固有振型的变化规律,分析比较了高跨比、梁上集中静荷载大小等因素对于固有频率的影响,并开展了相应的室内试验。结果表明:对于同一固支梁,固有频率随梁上静荷载的增大呈线性增大趋势;对于两端固支的3mm梁,初始挠度对固有频率的影响可达到17.3%;随着梁高跨比的减小,初始挠度对固有频率的影响逐渐增大。因此,实际的桥梁结构设计中,应充分考虑初始挠度对固有频率的影响。     

弹性圆拱在考虑几何非线性和初始缺陷情况下的动力稳定性分析

本文由虚功原理建立弹性圆拱的平衡方程,用有限差分法对非线性偏微分方程组进行求解(Park法对时间进行差分)。在考虑几何非线性和初始几何缺陷情况下对铰支、固支圆拱在均布突加阶跃荷载作用下的动力稳定性进行分析。结果表明:圆拱中心角的大小、边界条件及初始缺陷幅值都对圆拱失稳模态有影响。文中分析了直接、间接两种失稳形式。并给出了不同初始缺陷及边界条件下圆拱中心角对比值Pd/Pa(Pd为动力稳定临界值,Ps为静力稳定临界值)的影响。     

考虑恒载效应对梁静力反应影响的有限元分析

从非线性弹性理论出发,应用Hamilton原理,采用三次多项式作为位移函数给出了考虑恒载效应影响的一般形式的单元刚度矩阵,建立了考虑恒载效应影响时梁静力分析的有限元方法,为解决各种不同构造、荷载类型、刚度变化、边界条件下的实际结构考虑初始恒载影响提供了有效的工具。讨论了在分布和集中两种恒载类型条件下恒载大小、结构自身刚度（惯性矩、惯性半径、跨度）等因素对各类梁（简支梁、两端固支梁、悬臂梁）静力反应的影响,结果表明：梁自身的刚度会对恒载效应造成影响,结构越柔（约束越弱、或惯性矩越小、或跨度越大、或回转半径越小）,这种影响越大。验证了初始恒载的存在减小了梁在活载作用下的静力反应,并且这种影响与恒载的大小及结构自身的刚度有关。     

圆板在物体撞击下的非线性动力响应

本文在Von Kármán大位移的意义上,利用虚位移原理伽辽金方法建立了圆板在物体撞击下的非线性动力响应的控制微分方程,在研究响应问题时,考虑了冲击载荷与圆板位移响应之间的耦合影响,文中使用时间增量法和奇异摄动理论求解问题的控制方程,获得了固支圆板非线性动力响应的近似解,并且求解了具体算例,绘出了圆板位移、应力响应曲线以及冲击力随时间的变化曲线。     

圆板在物体撞击下的非线性动力响应

本文在Von Kármán大位移的意义上,利用虚位移原理伽辽金方法建立了圆板在物体撞击下的非线性动力响应的控制微分方程,在研究响应问题时,考虑了冲击载荷与圆板位移响应之间的耦合影响,文中使用时间增量法和奇异摄动理论求解问题的控制方程,获得了固支圆板非线性动力响应的近似解,并且求解了具体算例,绘出了圆板位移、应力响应曲线以及冲击力随时间的变化曲线。     

荷载横向变位下变截面连续箱梁剪滞分析的有限梁段法

为研究荷载横向作用位置变化对箱梁剪滞效应的影响,对箱梁顶、底板、悬臂板分别设置了不同的剪滞纵向位移差函数;假定纵向翘曲位移沿横向分布为k次抛物线,并考虑剪滞和剪切双重效应的影响,通过能量变分法推导出了荷载横向变位时梁段单元的平衡控制微分方程组及其闭合解;提出了能对工程中常见的变截面连续箱梁剪滞效应进行分析的有限梁段法。该方法计算结果与有限元模型、已有模型试验结果的最大误差在5.95%~9.74%之间,两种工况下计算结果的叠加与有限元结果相对误差在0.07%~19.18%之间,均吻合良好,说明将基于有限梁段法的剪滞效应变分解和叠加原理用于求解复杂力状态下的剪滞效应是可行的。剪滞翘曲位移横向分布函数精度选择的研究结果表明:均布荷载分别作用于腹板顶部、顶板中心时,翘曲位移横向分布函数宜分别选用三次、二次抛物线。     

考虑恒载效应的拱形梁静力近似解

应用虚功原理,推导了考虑恒载效应影响时拱形梁在活载作用下的非线性微分方程,得到了方程的近似闭合解。根据方程的解,讨论了恒载大小及结构自身刚度(矢高、跨度、惯性矩及惯性半径等)不同因素在考虑恒载效应时对拱形梁静力特性的影响。通过与Takabatake得到的直梁解析解结果及作者在其他文献提出的有限元方法对拱形梁分析结果的比较,验证了本文非线性微分方程及其求解公式。结果表明,本文给出的非线性微分方程对于拱形梁和直线梁具有通用性,初始恒载的存在减小了拱形梁在活载作用下的静力反应,这种影响与恒载的大小及结构自身的刚度有关,对轻型结构的设计提出了一些建议。     

Deflection of sandwich plates in terms of corresponding Kirchhoff plate solutions

Summary This study presents exact relationships between the deflections of isotropic sandwich plates and their corresponding Kirchhoff plates. The governing equilibrium equations for the sandwich plates are derived on the basis of the Reissner-Mindlin shear deformation plate theory. The considered plates are either (i) simply supported, of general polygonal shape and under any transverse loading condition or (ii) simply supported and clamped circular plates under axisymmetric loading. As the relationships are exact under the assumptions used in the plate theories, one may obtain exact deflection solutions of sandwich plates if the Kirchhoff plate solutions are exact. The relationships should also be useful for the development of approximate formulas for plates with other shapes, boundary and loading conditions, and may serve to check numerical deflection values computed from sandwich plate analysis software.     

Variational analysis of gradient elastic flexural plates under static loading

Gradient elastic flexural Kirchhoff plates under static loading are considered. Their governing equation of equilibrium in terms of their lateral deflection is a sixth order partial differential equation instead of the fourth order one for the classical case. A variational formulation of the problem is established with the aid of the principle of virtual work and used to determine all possible boundary conditions, classical and non-classical ones. Two circular gradient elastic plates, clamped or simply supported at their boundaries, are analyzed analytically and the gradient effect on their static response is assessed in detail. A rectangular gradient elastic plate, simply supported at its boundaries, is also analyzed analytically and its rationally obtained boundary conditions are compared with the heuristically obtained ones in a previous publication of the authors. Finally, a plate with two opposite sides clamped experiencing cylindrical bending is also analyzed and its response compared against that for the cases of micropolar and couple-stress elasticity theories.     

Secondary buckling of a flat plate under uniaxial compression: Part 2. Analysis of clamped plate by F.E.M. and comparison with experiments

In Part 1, a theoretical analysis was used to study the secondary buckling of a simply supported plate. In Part 2, a clamped plate is analysed by the finite element method. The stability criterion for a non-linear post-buckling equilibrium state is evaluated by the sign of the determinant of the stiffness matrix. It should be noted that the secondary buckling loads of clamped plates are unexpectedly smaller than those of simply supported plates and are only one and a half times the primary buckling loads. In previous analyses, only a quarter segment of the plate was considered by assuming a stable equilibrium state with symmetrical mode. However, instability can also be predicted by considering the unsymmetrical mode over the whole plate. Results of experimental analysis of secondary instability of clamped square plates under uniaxial compression agreed with the numerical results.     

BENDING OF SECTORIAL LEVINSON PLATES: RELATIONSHIPS WITH CLASSICAL SOLUTIONS*

ABSTRACT

Relationships between bending solutions (deflections and stress resultants) of the Levinson third-order plate theory are presented in terms of the solutions of the classical plate theory for sectorial plates. These relationships are algebraic in nature, and can be readily used to determine the shear deformation theory (based on the Levinson plate theory) solutions of sectorial plates without going through the process of solving the governing equations. The sectorial plates considered have simply supported radial edges while the circular curved edges may be either simply supported, clamped, or free. Numerical results are presented to illustrate the usefulness of the developed relationships.

     

预应力条件下轴对称圆板的响应解

本文研究预应力条件下轴对称简支和固支圆板的瞬态响应,利用振型展开和Laplace变换给出解析表达式,提出简化型圆板的边界条件,利用Hankel变换和Laplace变换给出简捷的解,针对几种边界状态,比较了其频率和响应,同时讨论了几种常见的瞬态载荷的结构响应形式。     

Exact solutions for the stability of viscoelastic rectangular plate subjected to tangential follower force

An exact solution procedure is formulated for the stability analysis of viscoelastic rectangular plate with two opposite edges simply supported and other two edges clamped as well as the viscoelastic rectangular plate with one edge clamped and other three edges simply supported under the action of tangential follower force. Firstly, by assuming the transverse displacement (W) as independent functions which automatically satisfies the simply supported boundary conditions, the governing partial differential equation is reduced to an ordinary differential equation with variable coefficients. Then, by the normalized power series method and applying the boundary conditions yield the eigenvalue problem of finding the roots of a fourth-order characteristic determinant. The results show that the aspect ratio λ and the dimensionless delay time H have great effects on the types of instability and the critical loads of the viscoelastic plates.     

Axisymmetric buckling of transversely isotropic circular and annular plates

This paper presents a general solution of the three-dimensional governing equations for the axisymmetric buckling problem of transversely isotropic media. The solution is expressed by a displacement function that satisfies a homogeneous fourth-order partial differential equation. Using this general solution, the axisymmetric buckling of circular and annular plates is investigated and exact solutions are obtained for appropriate boundary conditions. Numerical results are considered for clamped and simply supported circular and annular plates in comparison with existent results.     

功能梯度中厚圆/环板轴对称弯曲问题的解析解

基于一阶剪切变形板理论，导出了热/机载荷作用下，位移形式的功能梯度 中厚圆/环板轴对称弯曲问题的控制方程，获得了问题的位移和内力的一般解析解. 作为特 例，分别研究了边缘径向固定和可动的夹紧和简支的4种实心功能梯度圆板，给出了它们的 解，并分析了热/机载荷作用下解的形态，讨论了横向剪切变形、材料梯度常数和边界条件， 对板的轴对称弯曲行为的影响.