用材料力学公式计算任意截面短梁正应力时的修正项研究

对材料力学中梁的弯曲应力公式增加一修正项,以反映短梁弯剪翘曲变形对应力分布的影响。提出一种根据短梁横截面边界形状及艾瑞应力函数求解应力修正项的方法,应用弹性力学空间问题的一般理论,通过应力平衡方程、应变相容方程及应力边界条件,建立了关于任意截面短梁的应力修正项及剪应力的基本方程。在所建立的基本方程基础上,导出了矩形截面和圆形截面短梁修正应力的具体计算公式,该修正应力与均布荷载大小及弹性模量与剪切模量之比均成正比,但与截面惯性矩成反比。数值算例表明,本文方法计算的应力与通用有限元软件ANSYS计算的结果吻合良好,从而验证了本文方法及其基本公式的正确性。     

The shape of the strongest column

The problem of determining that shape of column which has the largest critical buckling load is solved, assuming that the length and volume are given and that each cross section is convex. The strongest column has an equilateral triangle as cross section, and it is tapered along its length, being thickest in the middle and thinnest at its ends. Its buckling load is 61.2% larger than that of a circular cylinder. For columns all of whose cross sections are similar and of prescribed shape-not necessarily convex—the best tapering is found to increase the buckling load by one third over that of a uniform column. This result, which was independently obtained by H. F. Weinberger, is originally due to Clausen (1851). For a uniform column, triangularizing is shown to increase the buckling load by 20.9% over that of a circular cylinder. The results lead to isoperimetric inequalities for the buckling loads of arbitrary columns. The research reported in this paper has been sponsored by the Office of Naval Research under Contract No. (285) 46.     

Kovalevskaya弹性杆

应用Kirchhoff比拟讨论Kovalevskaya情况弹性细杆的平衡稳定性问题．导出Kirchhoff方程的解析积分．对于杆截面的主轴与Frenet坐标轴重合的无扭转杆的特殊情形作定性分析，讨论其平衡状态的稳定性与分岔．证明了判断受拉扭作用的圆截面直杆平衡稳定性的Greenhill公式也适用于Kovalevskaya情形的非圆截面杆．     

Stress bounds for bars in torsion

Upper bounds for the maximum shear stress in the St. Venant torsion problem are derived with the aid of the theory of subharmonic functions. The main result is a bound that is determined in a simple manner by the magnitude of the applied twisting moment and two parameters peculiar to the cross section: the radius of the largest circle contained in it and the minimum curvature of the curve that bounds it.

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Zusammenfassung Mit Hilfe subharmonischer Funktionen werden obere Grenzen in dem Torsionsproblem von St. Venant erhalten. Das Hauptergebnis ist eine Grenze, die auf einfache Weise vom Drehmoment und zwei nur vom Querschnitt abhängigen Parametern bestimmt ist, und zwar von dem Radius des grössten eingeschriebenen Kreises und von der Minimalkrümmung der Begrenzungskurve des Querschnitts.

     

Stability analysis of helical rod based on exact Cosserat model

Helical equilibrium of a thin elastic rod has practical backgrounds, such as DNA, fiber, sub-ocean cable, and oil-well drill string. Kirchhoff's kinetic analogy is an effective approach to the stability analysis of equilibrium of a thin elastic rod. The main hypotheses of Kirchhoff's theory without the extension of the centerline and the shear deformation of the cross section are not adoptable to real soft materials of biological fibers. In this paper, the dynamic equations of a rod with a circular cross section are established on the basis of the exact Cosserat model by considering the tension and the shear deformations. Euler's angles are applied as the attitude representation of the cross section. The deviation of the normal axis of the cross section from the tangent of the centerline is considered as the result of the shear deformation. Lyapunov's stability of the helical equilibrium is discussed in static category. Euler's critical values of axial force and torque are obtained. Lyapunov's and Euler's stability conditions in the space domain are the necessary conditions of Lyapunov's stability of the helical rod in the time domain.     

含双裂纹圆截面悬臂梁的损伤识别研究

裂纹的萌生和扩展直接影响构件的振动响应,对构件的安全可靠性具有重要影响．本文以圆截面悬臂梁为对象,结合转角模态振型和模态频率等高线,研究了一种双裂纹识别技术．首先,基于应力强度因子和卡氏定理推导了无裂纹梁单元和含裂纹梁单元的刚度矩阵;在此基础上,建立了含裂纹圆截面悬臂梁的有限元动力学方程;然后,结合裂纹对梁转角模态振型和模态频率的影响,提出了双裂纹识别策略．最后,通过算例讨论了双裂纹识别策略的可行性．结果表明,圆截面悬臂梁的模态转角在裂纹位置出现突变,裂纹深度越大转角突变值越大;将识别出的裂纹位置作为已知参数,通过模态频率等高线法,可以准确地识别出双裂纹的深度．     

A Hamiltonian State Space Approach for 3D Analysis of Circular Cantilevers

A 3D exact analysis of extension, torsion and bending of a cantilever of a circular cross section is studied with emphasis on the fixed-end effect. Through Hamiltonian variational formulation, the basic equations of elasticity in cylindrical coordinates and the boundary conditions of the problem are formulated into the state space setting in which the state vector comprises the displacement vector and the conjugate stress vector as the dual variables. Upon delineating the Hamiltonian characteristics of the system, 3D solutions for transversely isotropic circular cantilevers subjected to an axial force, a torque, terminal couples and transverse forces are determined, thereby, the fixed-end effects and applicability of the solutions of generalized plane strains and the elementary theory of bending of beams are evaluated.     

钢桁腹式混凝土组合箱梁的扭转效应分析

结合钢桁腹式混凝土组合箱梁的结构特点,基于薄壁箱梁扭转理论,推导出组合箱梁闭口断面的混凝土顶底板和换算钢腹板的扭转翘曲应力表达式,进而推导出组合箱梁约束扭转控制微分方程;利用初参数法求解微分方程,并分析出翘曲双力矩以及扭转翘曲正应力随梁跨的变化规律.通过有限元模拟分析,将有限元值和理论值进行比较,结果吻合良好.研究表明,翘曲双力矩在集中扭矩作用处达到最大值,并且衰减速度很快,使得该处箱梁截面的翘曲应力达到最大值,箱梁的翘曲双力矩在远离集中扭矩作用处几乎为零,翘曲正应力也几乎为零;有限元数值与理论数值的差值百分比在10％以内,说明本文建立的理论计算方法合理可行.     

非圆截面弹性细杆的螺旋线平衡及稳定性

本文研究端部受力和力矩作用，且存在初曲率和初扭率的非圆截面弹性细杆的螺旋线平衡及其稳定性。描述弹性细杆平衡状态的Kirchhoff方程存在与杆的螺旋线平衡状态相对应的特解。直杆和圆环杆为螺旋线状态的两种特例。文中分析了螺旋线的几何特性与作用力和力矩之间的相互关系，并导出螺旋线平衡的一次近似解析形式稳定性判据。分析表明，松弛状态下弹性杆可处于螺旋线状态，直杆只有在轴向压力的作用下才能保持螺旋线平衡。无初曲率和初扭率弹性杆的螺旋线平衡稳定性必要条件是杆截面绕副法线轴的抗弯刚度大于或等于绕法线轴的抗弯刚度。此条件也适用于带初扭率的圆环杆及更普遍情形。无初曲率和初扭率的圆截面杆的螺旋线平衡恒稳定。     

开口厚壁截面短构件的约束扭转1）

为了分析开口厚壁截面短构件的约束扭转问题,采用统一分析梁模型与有限节线法,对T形和L形厚壁截面短构件约束扭转时横截面的翘曲和应力分布情况等问题进行了分析研究.算例计算结果表明：开口厚壁截面短构件存在与其横截面形心位置不一致的扭转（弯曲）中心,构件在不过扭转中心的外力作用下会产生弯扭耦合变形,其横截面将产生不均匀翘曲,横截面上的翘曲正应力和扭转剪应力均呈非线性分布.     

On bounds for the torsional stiffness of shafts of varying circular cross section

We state upper and lower bound formulas for the torsional stiffness of shafts of varying circular cross section, in accordance with the classical Michell formulation of this problem, through use of the principles of minimum potential and complementary energy. The general results are used to obtain explicit first-approximation bounds which, for the limiting case of the cylindrical shaft, reproduce the known elementary exact results. It is conjectured that the first-approximation lower bound is significantly closer to the exact result than the first-approximation upper bound.A report on work supported by the Office of Naval Research, Washington, D.C.     

Transverse radiation pressure and torque exerted by a plane electromagnetic wave on a rotating circular cylinder of isotropic material

Relativistically correct expressions are derived for the time-averaged total force and torque acting on a rotating conducting circular cylinder, immersed in an incident plane electromagnetic wave. With the help of Maxwell's stress tensor analytical expressions can be obtained for the time-averaged force and torque.     

A screw dislocation near a circular nano-inhomogeneity in gradient elasticity

A screw dislocation outside an infinite cylindrical nano-inhomogeneity of circular cross section is considered within the isotropic theory of gradient elasticity. Fields of total displacements, elastic and plastic distortions, elastic strains and stresses are derived and analyzed in detail. In contrast with the case of classical elasticity, the gradient solutions are shown to possess no singularities at the dislocation line. Moreover, all stress components are continuous and smooth at the interface unlike the classical solution. As a result, the image force exerted on the dislocation due to the differences in elastic and gradient constants of the matrix and inhomogeneity, remains finite when the dislocation approaches the interface. The gradient solution demonstrates a non-classical size-effect in such a way that the stress level inside the inhomogeneity decreases with its size. The gradient and classical solutions coincide when the distances from the dislocation line and the interface exceed several atomic spacings.     

THREE-DIMENSIONAL ANALYSIS OF A MAGNETOELECTROELASTIC HALF-SPACE UNDER AXISYMMETRIC TEMPERATURE LOADING 1)

考虑力-电-磁-热等多场耦合作用, 基于线性理论给出了磁-电-弹性半空间在表面轴对称温度载荷作用下的热-磁-电-弹性分析, 并得到了问题的解析解. 利用Hankel 积分变换法求解了磁-电-弹性材料中的热传导及控制方程, 讨论了在磁-电-弹性半空间在边界表面上作用局部热载荷时的混合边值问题, 利用积分变换和积分方程技术, 通过在边界表面上施加应力自由及磁-电开路条件, 推导得到了磁-电-弹性半空间中位移、电势及磁势的积分形式的表达式. 获得了磁-电-弹性半空间中温度场的解析表达式并且给出了应力, 电位移和磁通量的解析解. 数值计算结果表明温度载荷对磁-电-弹性场的分布有显著影响. 当温度载荷作用的圆域半径增大时, 最大正应力发生的位置会远离半无限大体的边界; 反之当温度载荷作用的圆域半径减小时, 最大应力发生的位置会靠近半无限大体的边界. 电场和磁场在温度载荷作用的圆域内在边界表面附近有明显的强化, 而磁-电-弹性场强化区域的强化程度跟温度载荷的大小和作用区域大小相关. 本研究的相关结果对智能材料和结构在热载荷作用下的设计和制造具有指导意义.     

On Shear and Torsion Factors in the Theory of Linearly Elastic Rods

Lower bounds for the factors entering the standard notions of shear and torsion stiffness for a linearly elastic rod are established in a new and simple way. The proofs are based on the following criterion to identify the stiffness parameters entering rod theory: the rod’s stored-energy density per unit length expressed in terms of force and moment resultants should equal the stored-energy density per unit length expressed in terms of stress components of a Saint-Venant cylinder subject to either flexure or torsion, according to the case. It is shown that the shear factor is always greater than one, whatever the cross section, a fact that is customarily stated without proof in textbooks of structure mechanics; and that the torsion factor is also greater than one, except when the cross section is a circle or a circular annulus, a fact that is usually proved making use of Saint-Venant’s solution in terms of displacement components.     

磁-电-弹性半空间在轴对称热载荷作用下的三维问题研究

考虑力-电-磁-热等多场耦合作用, 基于线性理论给出了磁-电-弹性半空间在表面轴对称温度载荷作用下的热-磁-电-弹性分析, 并得到了问题的解析解. 利用Hankel 积分变换法求解了磁-电-弹性材料中的热传导及控制方程, 讨论了在磁-电-弹性半空间在边界表面上作用局部热载荷时的混合边值问题, 利用积分变换和积分方程技术, 通过在边界表面上施加应力自由及磁-电开路条件, 推导得到了磁-电-弹性半空间中位移、电势及磁势的积分形式的表达式. 获得了磁-电-弹性半空间中温度场的解析表达式并且给出了应力, 电位移和磁通量的解析解. 数值计算结果表明温度载荷对磁-电-弹性场的分布有显著影响. 当温度载荷作用的圆域半径增大时, 最大正应力发生的位置会远离半无限大体的边界; 反之当温度载荷作用的圆域半径减小时, 最大应力发生的位置会靠近半无限大体的边界. 电场和磁场在温度载荷作用的圆域内在边界表面附近有明显的强化, 而磁-电-弹性场强化区域的强化程度跟温度载荷的大小和作用区域大小相关. 本研究的相关结果对智能材料和结构在热载荷作用下的设计和制造具有指导意义.      

An analytical theory of heated duct flows in supersonic combustors

One-dimensional analytical theory is developed for supersonic duct flow with variation of cross section, wall friction, heat addition, and relations between the inlet and outlet flow parameters are obtained. By introducing a selfsimilar parameter, effects of heat releasing, wall friction, and change in cross section area on the flow can be normalized and a self-similar solution of the flow equations can be found. Based on the result of self-similar solution, the sufficient and necessary condition for the occurrence of thermal choking is derived. A relation of the maximum heat addition leading to thermal choking of the duct flow is derived as functions of area ratio, wall friction, and mass addition, which is an extension of the classic Rayleigh flow theory, where the effects of wall friction and mass addition are not considered. The present work is expected to provide fundamentals for developing an integral analytical theory for ramjets and scramjets.     

Propagation of a long wave in a curved duct (I) basic analysis of long wave propagation in a curved duct with variable cross section

The propagation of a long wave in a three-dimensional curved duct with variable cross section is studied in this paper. It is shown that a three-dimensional Helmholtz equation can be decomposed into a two-dimensional Laplace (or Poisson) equation and a one-dimensional Webster equation by the curvilinear orthogonal coordinate system, non-dimensionization of reduced wave equation and regular perturbation with small parameterka, wherek is the wave number anda is the characteristic radius of the duct. The influences of the duct's geometric parameters (the area variation of the cross section, the curvature and torsion of the central line) on the asymptotic expansion of the solution are analysed. It is concluded that the effects of the variation of the cross sectional area first appear in the first term of the asymptotic expansion, and when the cross section shape has certain symmetric properties, the effects of the curvature and torsion of the central line first appear in the third and the fourth terms, respectively. An example of long wave propagation in a curved circular duct is also given at the end of this paper.     

The radar cross section of the semi-infinite elliptic cone

In order to determine the radar cross section of a semi-infinite elliptic cone first the solution of the pertinent canonical boundary value problem in sphero-conal coordinates is derived. For this purpose one has to solve a two-parameter eigenvalue problem with two coupled Lamé differential equations. The exact nose-on radar cross section of the semi-infinite elliptic cone has been evaluated for two orthogonal polarizations of the incident plane wave. In the degenerate case of a circular cone the known formula first deduced by Hansen and Schiff is obtained. The theoretical results also hold for the case of a plane angular sector which is another degenerate case of the elliptic cone.     

Pressure-flow relationships for steady flow through an eccentric double circular tube

Pressure-flow relationships are derived for the steady flow through an eccentric double circular tube whose cross section takes the form of an annulus bounded by two circles which are in general not concentric. The velocity distribution can be obtained by solving a two-dimensional Poisson equation with the use of conformal mapping. In a case when the two circles touch, a mapping function of different type is required to solve the equation. The relationships thus derived are reduced to the well-known formula for the limit of concentric circles. It is found that when the region between touching circles becomes sufficiently thin, the tube conductance behaves like the cube of the difference of radii of the two circles.